Algebra Flashcards: Compose Two Functions

Study Compose Two Functions in Algebra with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

Algebra

Compose Two Functions

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QUESTION

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Find $(f\circ g)(x)$ if $f(x)=x+1$ and $g(x)=\frac{1}{x}$ .

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ANSWER

$(f\circ g)(x)=\frac{1}{x}+1$ . Apply $f(x)=x+1$ to $g(x)=\frac{1}{x}$ .

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This deck focuses on Compose Two Functions, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra.

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Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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Flashcard 1: Find $(f\circ g)(x)$ if $f(x)=x+1$ and $g(x)=\frac{1}{x}$ .

Answer: $(f\circ g)(x)=\frac{1}{x}+1$ . Apply $f(x)=x+1$ to $g(x)=\frac{1}{x}$ .

Flashcard 2: Identify the inner function in $f(g(x))$ .

Answer: The inner function is $g(x)$ . The inner function is evaluated first in composition.

Flashcard 3: Find $(g\circ f)(x)$ if $f(x)=\sqrt{x}$ and $g(x)=x+9$ .

Answer: $(g\circ f)(x)=\sqrt{x}+9$ . Add 9 after taking the square root of $x$ .

Flashcard 4: What is the domain of $(g\circ f)(x)$ if $f(x)=2x-1$ and $g(x)=\sqrt{x}$ ?

Answer: Domain: $x\ge \frac{1}{2}$ . Need $2x-1\ge 0$ , so $x\ge \frac{1}{2}$ .

Flashcard 5: Find $(f\circ g)(x)$ if $f(x)=x^2-1$ and $g(x)=x+3$ .

Answer: $(f\circ g)(x)=(x+3)^2-1$ . Substitute $g(x)=x+3$ into $f(x)=x^2-1$ .

Flashcard 6: Find $(f\circ g)(x)$ if $f(x)=\frac{1}{x}$ and $g(x)=x-4$ .

Answer: $(f\circ g)(x)=\frac{1}{x-4}$ . Substitute $g(x)=x-4$ into $f(x)=\frac{1}{x}$ .

Flashcard 7: What is the domain of $(g\circ f)(x)$ if $f(x)=\frac{1}{x}$ and $g(x)=x-4$ ?

Answer: Domain: $x\ne 0$ . Need $x\ne 0$ for $f(x)=\frac{1}{x}$ to be defined.

Flashcard 8: State the associative property for composition using $f$ , $g$ , and $h$ .

Answer: $(f\circ g)\circ h=f\circ(g\circ h)$ . Composition groups the same way regardless of parentheses.

Flashcard 9: Find $(g\circ f)(x)$ if $f(x)=3x$ and $g(x)=\frac{x}{2}+5$ .

Answer: $(g\circ f)(x)=\frac{3}{2}x+5$ . Apply $g$ to $f(x)=3x$ to get $\frac{3x}{2}+5$ .

Flashcard 10: What does the notation $(f\circ g)(x)$ mean in words?

Answer: Apply $g$ first, then apply $f$ to the result. In composition, the inner function executes before the outer function.

Flashcard 11: Find $(f\circ g)(x)$ if $f(x)=x-10$ and $g(x)=x^2+1$ .

Answer: $(f\circ g)(x)=x^2-9$ . Apply $f(x)=x-10$ to $g(x)=x^2+1$ .

Flashcard 12: What is $(g\circ f)(2)$ if $f(x)=x-5$ and $g(x)=3x$ ?

Answer: $(g\circ f)(2)=-9$ . $f(2)=-3$ , then $g(-3)=-9$ .

Flashcard 13: Find $(f\circ g)(x)$ if $f(x)=x+1$ and $g(x)=\frac{1}{x}$ .

Answer: $(f\circ g)(x)=\frac{1}{x}+1$ . Apply $f(x)=x+1$ to $g(x)=\frac{1}{x}$ .

Flashcard 14: What is the domain of $(f\circ g)(x)$ if $f(x)=\frac{1}{x}$ and $g(x)=x-4$ ?

Answer: Domain: $x\ne 4$ . Need $x-4\ne 0$ , so $x\ne 4$ .

Flashcard 15: Find $(f\circ g)(x)$ if $f(x)=x-10$ and $g(x)=x^2+1$ .

Answer: $(f\circ g)(x)=x^2-9$ . Apply $f(x)=x-10$ to $g(x)=x^2+1$ .

Flashcard 16: Find $(g\circ f)(x)$ if $f(x)=2x-1$ and $g(x)=\sqrt{x}$ .

Answer: $(g\circ f)(x)=\sqrt{2x-1}$ . Apply $g(x)=\sqrt{x}$ to $f(x)=2x-1$ .

Flashcard 17: What is $(g\circ f)(-1)$ if $f(x)=x^2$ and $g(x)=x+4$ ?

Answer: $(g\circ f)(-1)=5$ . $f(-1)=1$ , then $g(1)=5$ .

Flashcard 18: What is $(f\circ g)(2)$ if $f(x)=x-5$ and $g(x)=3x$ ?

Answer: $(f\circ g)(2)=1$ . $g(2)=6$ , then $f(6)=6-5=1$ .

Flashcard 19: What is the domain of $(f\circ g)(x)$ if $f(x)=\sqrt{x}$ and $g(x)=x+9$ ?

Answer: Domain: $x\ge -9$ . Need $x+9\ge 0$ , so $x\ge -9$ .

Flashcard 20: What is the domain of $(g\circ f)(x)$ if $f(x)=2x-1$ and $g(x)=\sqrt{x}$ ?

Answer: Domain: $x\ge \frac{1}{2}$ . Need $2x-1\ge 0$ , so $x\ge \frac{1}{2}$ .

Flashcard 21: Identify the inner function in $f(g(x))$ .

Answer: The inner function is $g(x)$ . The inner function is evaluated first in composition.

Flashcard 22: What is the domain of $(g\circ f)(x)$ if $f(x)=\sqrt{x}$ and $g(x)=x+9$ ?

Answer: Domain: $x\ge 0$ . Need $x\ge 0$ for the square root in $f(x)$ .

Flashcard 23: Find $(g\circ f)(x)$ if $f(x)=x-10$ and $g(x)=x^2+1$ .

Answer: $(g\circ f)(x)=(x-10)^2+1$ . Apply $g(x)=x^2+1$ to $f(x)=x-10$ .

Flashcard 24: Find $(g\circ f)(x)$ if $f(x)=x-10$ and $g(x)=x^2+1$ .

Answer: $(g\circ f)(x)=(x-10)^2+1$ . Apply $g(x)=x^2+1$ to $f(x)=x-10$ .

Flashcard 25: Find $(f\circ g)(x)$ if $f(x)=2x-1$ and $g(x)=\sqrt{x}$ .

Answer: $(f\circ g)(x)=2\sqrt{x}-1$ . Apply $f(x)=2x-1$ to $g(x)=\sqrt{x}$ .

Flashcard 26: What is $(g\circ f)(x)$ if $f(x)=x^3$ and $g(x)=2x$ ?

Answer: $(g\circ f)(x)=2x^3$ . Apply $g(x)=2x$ to $f(x)=x^3$ to get $2x^3$ .

Flashcard 27: What is $(f\circ g)(2)$ if $f(x)=x-5$ and $g(x)=3x$ ?

Answer: $(f\circ g)(2)=1$ . $g(2)=6$ , then $f(6)=6-5=1$ .

Flashcard 28: What is $(f\circ g)(-1)$ if $f(x)=x^2$ and $g(x)=x+4$ ?

Answer: $(f\circ g)(-1)=9$ . $g(-1)=3$ , then $f(3)=9$ .

Flashcard 29: Find $(f\circ g)(x)$ if $f(x)=3x$ and $g(x)=\frac{x}{2}+5$ .

Answer: $(f\circ g)(x)=\frac{3}{2}x+15$ . Apply $f(x)=3x$ to $g(x)=\frac{x}{2}+5$ .

Flashcard 30: What is the meaning of $T(h(t))$ if $T$ depends on height and $h$ depends on time?

Answer: Temperature as a function of time at the balloon's height. Composition links time to temperature through height dependency.

Flashcard 31: What is the domain of $(f\circ g)(x)$ if $f(x)=2x-1$ and $g(x)=\sqrt{x}$ ?

Answer: Domain: $x\ge 0$ . Need $x\ge 0$ for the square root in $g(x)$ .

Flashcard 32: What is $(f\circ g)(x)$ if $f(x)=x^3$ and $g(x)=2x$ ?

Answer: $(f\circ g)(x)=8x^3$ . Apply $f(x)=x^3$ to $g(x)=2x$ to get $(2x)^3=8x^3$ .

Flashcard 33: Identify whether composition is commutative: Is $(f\circ g)(x)=(g\circ f)(x)$ always true?

Answer: No, composition is not commutative in general. Function composition depends on order of application.

Flashcard 34: What is the domain of $(f\circ g)$ described as a set condition?

Answer: All $x$ in domain of $g$ with $g(x)$ in domain of $f$ . Only $x$ values where both functions are defined work.

Flashcard 35: What is the key difference between $(f\circ g)(x)$ and $(g\circ f)(x)$ ?

Answer: They reverse the order of application and can give different results. Order matters in composition; $f\circ g \ne g\circ f$ generally.

Flashcard 36: Find $(g\circ f)(x)$ if $f(x)=\frac{1}{x}$ and $g(x)=x-4$ .

Answer: $(g\circ f)(x)=\frac{1}{x}-4$ . Apply $g$ to $f(x)=\frac{1}{x}$ to get $\frac{1}{x}-4$ .

Flashcard 37: Find $(f\circ g)(x)$ if $f(x)=|x|$ and $g(x)=x-7$ .

Answer: $(f\circ g)(x)=|x-7|$ . Apply absolute value to $g(x)=x-7$ .

Flashcard 38: What is the definition of the composition $(g\circ f)(x)$ ?

Answer: $(g\circ f)(x)=g(f(x))$ . This reverses the order: $g$ applies to $f$ 's output.

Flashcard 39: Find $(f\circ g)(x)$ if $f(x)=x^2-1$ and $g(x)=x+3$ .

Answer: $(f\circ g)(x)=(x+3)^2-1$ . Substitute $g(x)=x+3$ into $f(x)=x^2-1$ .

Flashcard 40: Find and correct the notation error: Writing $(f\circ g)(x)=g(f(x))$ is incorrect; what is correct?

Answer: Correct: $(f\circ g)(x)=f(g(x))$ . The order in composition notation is reversed from evaluation order.

Flashcard 41: What is the definition of the composition $(f\circ g)(x)$ in terms of $f$ and $g$ ?

Answer: $(f\circ g)(x)=f(g(x))$ . The composition applies $f$ to the output of $g(x)$ .

Flashcard 42: What does $(f\circ g)(a)$ mean when $a$ is a number?

Answer: Compute $g(a)$ , then compute $f(g(a))$ . Evaluate inner function first, then outer function.

Flashcard 43: What does $(f\circ g)(a)$ mean when $a$ is a number?

Answer: Compute $g(a)$ , then compute $f(g(a))$ . Evaluate inner function first, then outer function.

Flashcard 44: What is the identity function $I(x)$ used for in composition?

Answer: $I(x)=x$ , and $(f\circ I)(x)=(I\circ f)(x)=f(x)$ . The identity function leaves any function unchanged in composition.

Flashcard 45: Choose the correct setup: If $C(n)$ is cost per item from items $n$ , what is cost as a function of time $t$ ?

Answer: $C(n(t))$ . Composition links time to cost through item quantity.

Flashcard 46: What does the notation $(f\circ g)(x)$ mean in words?

Answer: Apply $g$ first, then apply $f$ to the result. In composition, the inner function executes before the outer function.

Flashcard 47: What is $(g\circ f)(2)$ if $f(x)=x-5$ and $g(x)=3x$ ?

Answer: $(g\circ f)(2)=-9$ . $f(2)=-3$ , then $g(-3)=-9$ .

Flashcard 48: Identify the composition: If $f(x)=2x+3$ and $g(x)=x^2$ , what is $(g\circ f)(x)$ ?

Answer: $(g\circ f)(x)=(2x+3)^2$ . Substitute $f(x)=2x+3$ into $g(x)=x^2$ to get $(2x+3)^2$ .

Flashcard 49: Find $(f\circ g)(x)$ if $f(x)=\sqrt{x}$ and $g(x)=x+9$ .

Answer: $(f\circ g)(x)=\sqrt{x+9}$ . Substitute $g(x)=x+9$ into $f(x)=\sqrt{x}$ .

Flashcard 50: Identify the composition: If $f(x)=2x+3$ and $g(x)=x^2$ , what is $(f\circ g)(x)$ ?

Answer: $(f\circ g)(x)=2x^2+3$ . Substitute $g(x)=x^2$ into $f(x)=2x+3$ to get $2x^2+3$ .

Flashcard 51: Find $(f\circ g)(x)$ if $f(x)=\frac{1}{x}$ and $g(x)=x-4$ .

Answer: $(f\circ g)(x)=\frac{1}{x-4}$ . Substitute $g(x)=x-4$ into $f(x)=\frac{1}{x}$ .

Flashcard 52: Choose the correct setup: If $C(n)$ is cost per item from items $n$ , what is cost as a function of time $t$ ?

Answer: $C(n(t))$ . Composition links time to cost through item quantity.

Flashcard 53: Identify the correct composition: If $f(x)=x-2$ and $g(x)=x^2$ , what is $(f\circ g)(x)$ ?

Answer: $(f\circ g)(x)=x^2-2$ . Apply $f(x)=x-2$ to $g(x)=x^2$ .

Flashcard 54: What is the domain of $(g\circ f)(x)$ if $f(x)=x+1$ and $g(x)=\frac{1}{x}$ ?

Answer: Domain: $x\ne -1$ . Need $x+1\ne 0$ , so $x\ne -1$ .

Flashcard 55: What is $(f\circ g)(-1)$ if $f(x)=x^2$ and $g(x)=x+4$ ?

Answer: $(f\circ g)(-1)=9$ . $g(-1)=3$ , then $f(3)=9$ .

Flashcard 56: Find $(g\circ f)(x)$ if $f(x)=|x|$ and $g(x)=x-7$ .

Answer: $(g\circ f)(x)=|x|-7$ . Subtract 7 from the absolute value of $x$ .

Flashcard 57: Find $(g\circ f)(x)$ if $f(x)=2x-1$ and $g(x)=\sqrt{x}$ .

Answer: $(g\circ f)(x)=\sqrt{2x-1}$ . Apply $g(x)=\sqrt{x}$ to $f(x)=2x-1$ .

Flashcard 58: State the associative property for composition using $f$ , $g$ , and $h$ .

Answer: $(f\circ g)\circ h=f\circ(g\circ h)$ . Composition groups the same way regardless of parentheses.

Flashcard 59: What must be true about domains to make $(f\circ g)(x)$ well-defined?

Answer: The outputs of $g$ must be in the domain of $f$ . The range of $g$ must overlap with the domain of $f$ .

Flashcard 60: What is the domain of $(f\circ g)(x)$ if $f(x)=x+1$ and $g(x)=\frac{1}{x}$ ?

Answer: Domain: $x\ne 0$ . Need $x\ne 0$ for $g(x)=\frac{1}{x}$ to be defined.

Flashcard 61: Find $(g\circ f)(x)$ if $f(x)=x+1$ and $g(x)=\frac{1}{x}$ .

Answer: $(g\circ f)(x)=\frac{1}{x+1}$ . Apply $g(x)=\frac{1}{x}$ to $f(x)=x+1$ .

Flashcard 62: Identify the correct composition: If $f(x)=x-2$ and $g(x)=x^2$ , what is $(g\circ f)(x)$ ?

Answer: $(g\circ f)(x)=(x-2)^2$ . Apply $g(x)=x^2$ to $f(x)=x-2$ .

Flashcard 63: Find $(f\circ g)(x)$ if $f(x)=\sqrt{x}$ and $g(x)=x+9$ .

Answer: $(f\circ g)(x)=\sqrt{x+9}$ . Substitute $g(x)=x+9$ into $f(x)=\sqrt{x}$ .

Flashcard 64: Find $(g\circ f)(x)$ if $f(x)=\sqrt{x}$ and $g(x)=x+9$ .

Answer: $(g\circ f)(x)=\sqrt{x}+9$ . Add 9 after taking the square root of $x$ .

Flashcard 65: Which function is evaluated first in $(f\circ g)(x)$ : $f$ or $g$ ?

Answer: $g$ is evaluated first. Reading right to left, $g$ is the inner function in $(f\circ g)(x)$ .

Flashcard 66: Find $(g\circ f)(x)$ if $f(x)=x^2-1$ and $g(x)=x+3$ .

Answer: $(g\circ f)(x)=x^2+2$ . Add 3 to $f(x)=x^2-1$ to get $x^2+2$ .

Flashcard 67: What is a common notation for composing $f$ with $g$ besides $(f\circ g)(x)$ ?

Answer: $f(g(x))$ . This notation directly shows the nested function evaluation.

Flashcard 68: What is the domain of $(g\circ f)(x)$ if $f(x)=\frac{1}{x}$ and $g(x)=x-4$ ?

Answer: Domain: $x\ne 0$ . Need $x\ne 0$ for $f(x)=\frac{1}{x}$ to be defined.

Flashcard 69: Find and correct the notation error: Writing $(f\circ g)(x)=g(f(x))$ is incorrect; what is correct?

Answer: Correct: $(f\circ g)(x)=f(g(x))$ . The order in composition notation is reversed from evaluation order.

Flashcard 70: What is the domain of $(f\circ g)(x)$ if $f(x)=x+1$ and $g(x)=\frac{1}{x}$ ?

Answer: Domain: $x\ne 0$ . Need $x\ne 0$ for $g(x)=\frac{1}{x}$ to be defined.

Flashcard 71: Simplify $(f\circ g)(x)$ if $f(x)=x^2-1$ and $g(x)=x+3$ .

Answer: $(f\circ g)(x)=x^2+6x+8$ . Expand $(x+3)^2-1=x^2+6x+9-1=x^2+6x+8$ .

Algebra Flashcards: Compose Two Functions

Compose Two Functions

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Flashcard 1: Find (f∘g)(x)(f\circ g)(x)(f∘g)(x) if f(x)=x+1f(x)=x+1f(x)=x+1 and g(x)=1xg(x)=\frac{1}{x}g(x)=x1​.

Flashcard 2: Identify the inner function in f(g(x))f(g(x))f(g(x)).

Flashcard 3: Find (g∘f)(x)(g\circ f)(x)(g∘f)(x) if f(x)=xf(x)=\sqrt{x}f(x)=x​ and g(x)=x+9g(x)=x+9g(x)=x+9.

Flashcard 4: What is the domain of (g∘f)(x)(g\circ f)(x)(g∘f)(x) if f(x)=2x−1f(x)=2x-1f(x)=2x−1 and g(x)=xg(x)=\sqrt{x}g(x)=x​?

Flashcard 5: Find (f∘g)(x)(f\circ g)(x)(f∘g)(x) if f(x)=x2−1f(x)=x^2-1f(x)=x2−1 and g(x)=x+3g(x)=x+3g(x)=x+3.

Flashcard 6: Find (f∘g)(x)(f\circ g)(x)(f∘g)(x) if f(x)=1xf(x)=\frac{1}{x}f(x)=x1​ and g(x)=x−4g(x)=x-4g(x)=x−4.

Flashcard 7: What is the domain of (g∘f)(x)(g\circ f)(x)(g∘f)(x) if f(x)=1xf(x)=\frac{1}{x}f(x)=x1​ and g(x)=x−4g(x)=x-4g(x)=x−4?

Flashcard 8: State the associative property for composition using fff, ggg, and hhh.

Flashcard 9: Find (g∘f)(x)(g\circ f)(x)(g∘f)(x) if f(x)=3xf(x)=3xf(x)=3x and g(x)=x2+5g(x)=\frac{x}{2}+5g(x)=2x​+5.

Flashcard 10: What does the notation (f∘g)(x)(f\circ g)(x)(f∘g)(x) mean in words?

Flashcard 11: Find (f∘g)(x)(f\circ g)(x)(f∘g)(x) if f(x)=x−10f(x)=x-10f(x)=x−10 and g(x)=x2+1g(x)=x^2+1g(x)=x2+1.

Flashcard 12: What is (g∘f)(2)(g\circ f)(2)(g∘f)(2) if f(x)=x−5f(x)=x-5f(x)=x−5 and g(x)=3xg(x)=3xg(x)=3x?

Flashcard 13: Find (f∘g)(x)(f\circ g)(x)(f∘g)(x) if f(x)=x+1f(x)=x+1f(x)=x+1 and g(x)=1xg(x)=\frac{1}{x}g(x)=x1​.

Flashcard 14: What is the domain of (f∘g)(x)(f\circ g)(x)(f∘g)(x) if f(x)=1xf(x)=\frac{1}{x}f(x)=x1​ and g(x)=x−4g(x)=x-4g(x)=x−4?

Flashcard 15: Find (f∘g)(x)(f\circ g)(x)(f∘g)(x) if f(x)=x−10f(x)=x-10f(x)=x−10 and g(x)=x2+1g(x)=x^2+1g(x)=x2+1.

Flashcard 16: Find (g∘f)(x)(g\circ f)(x)(g∘f)(x) if f(x)=2x−1f(x)=2x-1f(x)=2x−1 and g(x)=xg(x)=\sqrt{x}g(x)=x​.

Flashcard 17: What is (g∘f)(−1)(g\circ f)(-1)(g∘f)(−1) if f(x)=x2f(x)=x^2f(x)=x2 and g(x)=x+4g(x)=x+4g(x)=x+4?

Flashcard 18: What is (f∘g)(2)(f\circ g)(2)(f∘g)(2) if f(x)=x−5f(x)=x-5f(x)=x−5 and g(x)=3xg(x)=3xg(x)=3x?

Flashcard 19: What is the domain of (f∘g)(x)(f\circ g)(x)(f∘g)(x) if f(x)=xf(x)=\sqrt{x}f(x)=x​ and g(x)=x+9g(x)=x+9g(x)=x+9?

Flashcard 20: What is the domain of (g∘f)(x)(g\circ f)(x)(g∘f)(x) if f(x)=2x−1f(x)=2x-1f(x)=2x−1 and g(x)=xg(x)=\sqrt{x}g(x)=x​?

Flashcard 21: Identify the inner function in f(g(x))f(g(x))f(g(x)).

Flashcard 22: What is the domain of (g∘f)(x)(g\circ f)(x)(g∘f)(x) if f(x)=xf(x)=\sqrt{x}f(x)=x​ and g(x)=x+9g(x)=x+9g(x)=x+9?

Flashcard 23: Find (g∘f)(x)(g\circ f)(x)(g∘f)(x) if f(x)=x−10f(x)=x-10f(x)=x−10 and g(x)=x2+1g(x)=x^2+1g(x)=x2+1.

Flashcard 24: Find (g∘f)(x)(g\circ f)(x)(g∘f)(x) if f(x)=x−10f(x)=x-10f(x)=x−10 and g(x)=x2+1g(x)=x^2+1g(x)=x2+1.

Flashcard 25: Find (f∘g)(x)(f\circ g)(x)(f∘g)(x) if f(x)=2x−1f(x)=2x-1f(x)=2x−1 and g(x)=xg(x)=\sqrt{x}g(x)=x​.

Flashcard 26: What is (g∘f)(x)(g\circ f)(x)(g∘f)(x) if f(x)=x3f(x)=x^3f(x)=x3 and g(x)=2xg(x)=2xg(x)=2x?

Flashcard 27: What is (f∘g)(2)(f\circ g)(2)(f∘g)(2) if f(x)=x−5f(x)=x-5f(x)=x−5 and g(x)=3xg(x)=3xg(x)=3x?

Flashcard 28: What is (f∘g)(−1)(f\circ g)(-1)(f∘g)(−1) if f(x)=x2f(x)=x^2f(x)=x2 and g(x)=x+4g(x)=x+4g(x)=x+4?

Flashcard 29: Find (f∘g)(x)(f\circ g)(x)(f∘g)(x) if f(x)=3xf(x)=3xf(x)=3x and g(x)=x2+5g(x)=\frac{x}{2}+5g(x)=2x​+5.

Flashcard 30: What is the meaning of T(h(t))T(h(t))T(h(t)) if TTT depends on height and hhh depends on time?

Flashcard 31: What is the domain of (f∘g)(x)(f\circ g)(x)(f∘g)(x) if f(x)=2x−1f(x)=2x-1f(x)=2x−1 and g(x)=xg(x)=\sqrt{x}g(x)=x​?

Flashcard 32: What is (f∘g)(x)(f\circ g)(x)(f∘g)(x) if f(x)=x3f(x)=x^3f(x)=x3 and g(x)=2xg(x)=2xg(x)=2x?

Flashcard 33: Identify whether composition is commutative: Is (f∘g)(x)=(g∘f)(x)(f\circ g)(x)=(g\circ f)(x)(f∘g)(x)=(g∘f)(x) always true?

Flashcard 34: What is the domain of (f∘g)(f\circ g)(f∘g) described as a set condition?

Flashcard 35: What is the key difference between (f∘g)(x)(f\circ g)(x)(f∘g)(x) and (g∘f)(x)(g\circ f)(x)(g∘f)(x)?

Flashcard 36: Find (g∘f)(x)(g\circ f)(x)(g∘f)(x) if f(x)=1xf(x)=\frac{1}{x}f(x)=x1​ and g(x)=x−4g(x)=x-4g(x)=x−4.

Flashcard 37: Find (f∘g)(x)(f\circ g)(x)(f∘g)(x) if f(x)=∣x∣f(x)=|x|f(x)=∣x∣ and g(x)=x−7g(x)=x-7g(x)=x−7.

Flashcard 38: What is the definition of the composition (g∘f)(x)(g\circ f)(x)(g∘f)(x)?

Flashcard 39: Find (f∘g)(x)(f\circ g)(x)(f∘g)(x) if f(x)=x2−1f(x)=x^2-1f(x)=x2−1 and g(x)=x+3g(x)=x+3g(x)=x+3.

Flashcard 40: Find and correct the notation error: Writing (f∘g)(x)=g(f(x))(f\circ g)(x)=g(f(x))(f∘g)(x)=g(f(x)) is incorrect; what is correct?

Flashcard 41: What is the definition of the composition (f∘g)(x)(f\circ g)(x)(f∘g)(x) in terms of fff and ggg?

Flashcard 42: What does (f∘g)(a)(f\circ g)(a)(f∘g)(a) mean when aaa is a number?

Flashcard 43: What does (f∘g)(a)(f\circ g)(a)(f∘g)(a) mean when aaa is a number?

Flashcard 44: What is the identity function I(x)I(x)I(x) used for in composition?

Flashcard 45: Choose the correct setup: If C(n)C(n)C(n) is cost per item from items nnn, what is cost as a function of time ttt?

Flashcard 46: What does the notation (f∘g)(x)(f\circ g)(x)(f∘g)(x) mean in words?

Flashcard 47: What is (g∘f)(2)(g\circ f)(2)(g∘f)(2) if f(x)=x−5f(x)=x-5f(x)=x−5 and g(x)=3xg(x)=3xg(x)=3x?

Flashcard 48: Identify the composition: If f(x)=2x+3f(x)=2x+3f(x)=2x+3 and g(x)=x2g(x)=x^2g(x)=x2, what is (g∘f)(x)(g\circ f)(x)(g∘f)(x)?

Flashcard 49: Find (f∘g)(x)(f\circ g)(x)(f∘g)(x) if f(x)=xf(x)=\sqrt{x}f(x)=x​ and g(x)=x+9g(x)=x+9g(x)=x+9.

Flashcard 50: Identify the composition: If f(x)=2x+3f(x)=2x+3f(x)=2x+3 and g(x)=x2g(x)=x^2g(x)=x2, what is (f∘g)(x)(f\circ g)(x)(f∘g)(x)?

Flashcard 51: Find (f∘g)(x)(f\circ g)(x)(f∘g)(x) if f(x)=1xf(x)=\frac{1}{x}f(x)=x1​ and g(x)=x−4g(x)=x-4g(x)=x−4.

Flashcard 52: Choose the correct setup: If C(n)C(n)C(n) is cost per item from items nnn, what is cost as a function of time ttt?

Flashcard 53: Identify the correct composition: If f(x)=x−2f(x)=x-2f(x)=x−2 and g(x)=x2g(x)=x^2g(x)=x2, what is (f∘g)(x)(f\circ g)(x)(f∘g)(x)?

Flashcard 54: What is the domain of (g∘f)(x)(g\circ f)(x)(g∘f)(x) if f(x)=x+1f(x)=x+1f(x)=x+1 and g(x)=1xg(x)=\frac{1}{x}g(x)=x1​?

Flashcard 55: What is (f∘g)(−1)(f\circ g)(-1)(f∘g)(−1) if f(x)=x2f(x)=x^2f(x)=x2 and g(x)=x+4g(x)=x+4g(x)=x+4?

Flashcard 56: Find (g∘f)(x)(g\circ f)(x)(g∘f)(x) if f(x)=∣x∣f(x)=|x|f(x)=∣x∣ and g(x)=x−7g(x)=x-7g(x)=x−7.

Flashcard 57: Find (g∘f)(x)(g\circ f)(x)(g∘f)(x) if f(x)=2x−1f(x)=2x-1f(x)=2x−1 and g(x)=xg(x)=\sqrt{x}g(x)=x​.

Flashcard 58: State the associative property for composition using fff, ggg, and hhh.

Flashcard 59: What must be true about domains to make (f∘g)(x)(f\circ g)(x)(f∘g)(x) well-defined?

Flashcard 60: What is the domain of (f∘g)(x)(f\circ g)(x)(f∘g)(x) if f(x)=x+1f(x)=x+1f(x)=x+1 and g(x)=1xg(x)=\frac{1}{x}g(x)=x1​?

Flashcard 61: Find (g∘f)(x)(g\circ f)(x)(g∘f)(x) if f(x)=x+1f(x)=x+1f(x)=x+1 and g(x)=1xg(x)=\frac{1}{x}g(x)=x1​.

Flashcard 62: Identify the correct composition: If f(x)=x−2f(x)=x-2f(x)=x−2 and g(x)=x2g(x)=x^2g(x)=x2, what is (g∘f)(x)(g\circ f)(x)(g∘f)(x)?

Flashcard 63: Find (f∘g)(x)(f\circ g)(x)(f∘g)(x) if f(x)=xf(x)=\sqrt{x}f(x)=x​ and g(x)=x+9g(x)=x+9g(x)=x+9.

Flashcard 64: Find (g∘f)(x)(g\circ f)(x)(g∘f)(x) if f(x)=xf(x)=\sqrt{x}f(x)=x​ and g(x)=x+9g(x)=x+9g(x)=x+9.

Flashcard 65: Which function is evaluated first in (f∘g)(x)(f\circ g)(x)(f∘g)(x): fff or ggg?

Flashcard 66: Find (g∘f)(x)(g\circ f)(x)(g∘f)(x) if f(x)=x2−1f(x)=x^2-1f(x)=x2−1 and g(x)=x+3g(x)=x+3g(x)=x+3.

Flashcard 67: What is a common notation for composing fff with ggg besides (f∘g)(x)(f\circ g)(x)(f∘g)(x)?

Flashcard 68: What is the domain of (g∘f)(x)(g\circ f)(x)(g∘f)(x) if f(x)=1xf(x)=\frac{1}{x}f(x)=x1​ and g(x)=x−4g(x)=x-4g(x)=x−4?

Flashcard 69: Find and correct the notation error: Writing (f∘g)(x)=g(f(x))(f\circ g)(x)=g(f(x))(f∘g)(x)=g(f(x)) is incorrect; what is correct?

Flashcard 70: What is the domain of (f∘g)(x)(f\circ g)(x)(f∘g)(x) if f(x)=x+1f(x)=x+1f(x)=x+1 and g(x)=1xg(x)=\frac{1}{x}g(x)=x1​?

Flashcard 71: Simplify (f∘g)(x)(f\circ g)(x)(f∘g)(x) if f(x)=x2−1f(x)=x^2-1f(x)=x2−1 and g(x)=x+3g(x)=x+3g(x)=x+3.