Compose Two Functions - Algebra
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Find $(f\circ g)(x)$ if $f(x)=x-10$ and $g(x)=x^2+1$.
Find $(f\circ g)(x)$ if $f(x)=x-10$ and $g(x)=x^2+1$.
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$(f\circ g)(x)=x^2-9$. Apply $f(x)=x-10$ to $g(x)=x^2+1$.
$(f\circ g)(x)=x^2-9$. Apply $f(x)=x-10$ to $g(x)=x^2+1$.
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What is the definition of the composition $(f\circ g)(x)$ in terms of $f$ and $g$?
What is the definition of the composition $(f\circ g)(x)$ in terms of $f$ and $g$?
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$(f\circ g)(x)=f(g(x))$. The composition applies $f$ to the output of $g(x)$.
$(f\circ g)(x)=f(g(x))$. The composition applies $f$ to the output of $g(x)$.
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What does the notation $(f\circ g)(x)$ mean in words?
What does the notation $(f\circ g)(x)$ mean in words?
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Apply $g$ first, then apply $f$ to the result. In composition, the inner function executes before the outer function.
Apply $g$ first, then apply $f$ to the result. In composition, the inner function executes before the outer function.
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Which function is evaluated first in $(f\circ g)(x)$: $f$ or $g$?
Which function is evaluated first in $(f\circ g)(x)$: $f$ or $g$?
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$g$ is evaluated first. Reading right to left, $g$ is the inner function in $(f\circ g)(x)$.
$g$ is evaluated first. Reading right to left, $g$ is the inner function in $(f\circ g)(x)$.
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What is the definition of the composition $(g\circ f)(x)$?
What is the definition of the composition $(g\circ f)(x)$?
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$(g\circ f)(x)=g(f(x))$. This reverses the order: $g$ applies to $f$'s output.
$(g\circ f)(x)=g(f(x))$. This reverses the order: $g$ applies to $f$'s output.
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What must be true about domains to make $(f\circ g)(x)$ well-defined?
What must be true about domains to make $(f\circ g)(x)$ well-defined?
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The outputs of $g$ must be in the domain of $f$. The range of $g$ must overlap with the domain of $f$.
The outputs of $g$ must be in the domain of $f$. The range of $g$ must overlap with the domain of $f$.
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What is the domain of $(f\circ g)$ described as a set condition?
What is the domain of $(f\circ g)$ described as a set condition?
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All $x$ in domain of $g$ with $g(x)$ in domain of $f$. Only $x$ values where both functions are defined work.
All $x$ in domain of $g$ with $g(x)$ in domain of $f$. Only $x$ values where both functions are defined work.
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What is the meaning of $T(h(t))$ if $T$ depends on height and $h$ depends on time?
What is the meaning of $T(h(t))$ if $T$ depends on height and $h$ depends on time?
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Temperature as a function of time at the balloon’s height. Composition links time to temperature through height dependency.
Temperature as a function of time at the balloon’s height. Composition links time to temperature through height dependency.
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What is the key difference between $(f\circ g)(x)$ and $(g\circ f)(x)$?
What is the key difference between $(f\circ g)(x)$ and $(g\circ f)(x)$?
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They reverse the order of application and can give different results. Order matters in composition; $f\circ g \ne g\circ f$ generally.
They reverse the order of application and can give different results. Order matters in composition; $f\circ g \ne g\circ f$ generally.
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What is the identity function $I(x)$ used for in composition?
What is the identity function $I(x)$ used for in composition?
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$I(x)=x$, and $(f\circ I)(x)=(I\circ f)(x)=f(x)$. The identity function leaves any function unchanged in composition.
$I(x)=x$, and $(f\circ I)(x)=(I\circ f)(x)=f(x)$. The identity function leaves any function unchanged in composition.
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What is a common notation for composing $f$ with $g$ besides $(f\circ g)(x)$?
What is a common notation for composing $f$ with $g$ besides $(f\circ g)(x)$?
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$f(g(x))$. This notation directly shows the nested function evaluation.
$f(g(x))$. This notation directly shows the nested function evaluation.
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Identify the composition: If $f(x)=2x+3$ and $g(x)=x^2$, what is $(f\circ g)(x)$?
Identify the composition: If $f(x)=2x+3$ and $g(x)=x^2$, what is $(f\circ g)(x)$?
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$(f\circ g)(x)=2x^2+3$. Substitute $g(x)=x^2$ into $f(x)=2x+3$ to get $2x^2+3$.
$(f\circ g)(x)=2x^2+3$. Substitute $g(x)=x^2$ into $f(x)=2x+3$ to get $2x^2+3$.
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Identify the composition: If $f(x)=2x+3$ and $g(x)=x^2$, what is $(g\circ f)(x)$?
Identify the composition: If $f(x)=2x+3$ and $g(x)=x^2$, what is $(g\circ f)(x)$?
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$(g\circ f)(x)=(2x+3)^2$. Substitute $f(x)=2x+3$ into $g(x)=x^2$ to get $(2x+3)^2$.
$(g\circ f)(x)=(2x+3)^2$. Substitute $f(x)=2x+3$ into $g(x)=x^2$ to get $(2x+3)^2$.
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What is $(f\circ g)(2)$ if $f(x)=x-5$ and $g(x)=3x$?
What is $(f\circ g)(2)$ if $f(x)=x-5$ and $g(x)=3x$?
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$(f\circ g)(2)=1$. $g(2)=6$, then $f(6)=6-5=1$.
$(f\circ g)(2)=1$. $g(2)=6$, then $f(6)=6-5=1$.
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What is $(g\circ f)(2)$ if $f(x)=x-5$ and $g(x)=3x$?
What is $(g\circ f)(2)$ if $f(x)=x-5$ and $g(x)=3x$?
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$(g\circ f)(2)=-9$. $f(2)=-3$, then $g(-3)=-9$.
$(g\circ f)(2)=-9$. $f(2)=-3$, then $g(-3)=-9$.
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Find $(f\circ g)(x)$ if $f(x)=\sqrt{x}$ and $g(x)=x+9$.
Find $(f\circ g)(x)$ if $f(x)=\sqrt{x}$ and $g(x)=x+9$.
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$(f\circ g)(x)=\sqrt{x+9}$. Substitute $g(x)=x+9$ into $f(x)=\sqrt{x}$.
$(f\circ g)(x)=\sqrt{x+9}$. Substitute $g(x)=x+9$ into $f(x)=\sqrt{x}$.
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Find $(g\circ f)(x)$ if $f(x)=\sqrt{x}$ and $g(x)=x+9$.
Find $(g\circ f)(x)$ if $f(x)=\sqrt{x}$ and $g(x)=x+9$.
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$(g\circ f)(x)=\sqrt{x}+9$. Add 9 after taking the square root of $x$.
$(g\circ f)(x)=\sqrt{x}+9$. Add 9 after taking the square root of $x$.
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What is the domain of $(f\circ g)(x)$ if $f(x)=\sqrt{x}$ and $g(x)=x+9$?
What is the domain of $(f\circ g)(x)$ if $f(x)=\sqrt{x}$ and $g(x)=x+9$?
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Domain: $x\ge -9$. Need $x+9\ge 0$, so $x\ge -9$.
Domain: $x\ge -9$. Need $x+9\ge 0$, so $x\ge -9$.
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What is the domain of $(f\circ g)(x)$ if $f(x)=\frac{1}{x}$ and $g(x)=x-4$?
What is the domain of $(f\circ g)(x)$ if $f(x)=\frac{1}{x}$ and $g(x)=x-4$?
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Domain: $x\ne 4$. Need $x-4\ne 0$, so $x\ne 4$.
Domain: $x\ne 4$. Need $x-4\ne 0$, so $x\ne 4$.
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Find $(g\circ f)(x)$ if $f(x)=\frac{1}{x}$ and $g(x)=x-4$.
Find $(g\circ f)(x)$ if $f(x)=\frac{1}{x}$ and $g(x)=x-4$.
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$(g\circ f)(x)=\frac{1}{x}-4$. Apply $g$ to $f(x)=\frac{1}{x}$ to get $\frac{1}{x}-4$.
$(g\circ f)(x)=\frac{1}{x}-4$. Apply $g$ to $f(x)=\frac{1}{x}$ to get $\frac{1}{x}-4$.
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What is the domain of $(g\circ f)(x)$ if $f(x)=\frac{1}{x}$ and $g(x)=x-4$?
What is the domain of $(g\circ f)(x)$ if $f(x)=\frac{1}{x}$ and $g(x)=x-4$?
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Domain: $x\ne 0$. Need $x\ne 0$ for $f(x)=\frac{1}{x}$ to be defined.
Domain: $x\ne 0$. Need $x\ne 0$ for $f(x)=\frac{1}{x}$ to be defined.
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Find $(f\circ g)(x)$ if $f(x)=|x|$ and $g(x)=x-7$.
Find $(f\circ g)(x)$ if $f(x)=|x|$ and $g(x)=x-7$.
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$(f\circ g)(x)=|x-7|$. Apply absolute value to $g(x)=x-7$.
$(f\circ g)(x)=|x-7|$. Apply absolute value to $g(x)=x-7$.
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Find $(g\circ f)(x)$ if $f(x)=|x|$ and $g(x)=x-7$.
Find $(g\circ f)(x)$ if $f(x)=|x|$ and $g(x)=x-7$.
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$(g\circ f)(x)=|x|-7$. Subtract 7 from the absolute value of $x$.
$(g\circ f)(x)=|x|-7$. Subtract 7 from the absolute value of $x$.
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Find $(f\circ g)(x)$ if $f(x)=x^2-1$ and $g(x)=x+3$.
Find $(f\circ g)(x)$ if $f(x)=x^2-1$ and $g(x)=x+3$.
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$(f\circ g)(x)=(x+3)^2-1$. Substitute $g(x)=x+3$ into $f(x)=x^2-1$.
$(f\circ g)(x)=(x+3)^2-1$. Substitute $g(x)=x+3$ into $f(x)=x^2-1$.
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Simplify $(f\circ g)(x)$ if $f(x)=x^2-1$ and $g(x)=x+3$.
Simplify $(f\circ g)(x)$ if $f(x)=x^2-1$ and $g(x)=x+3$.
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$(f\circ g)(x)=x^2+6x+8$. Expand $(x+3)^2-1=x^2+6x+9-1=x^2+6x+8$.
$(f\circ g)(x)=x^2+6x+8$. Expand $(x+3)^2-1=x^2+6x+9-1=x^2+6x+8$.
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Find $(g\circ f)(x)$ if $f(x)=x^2-1$ and $g(x)=x+3$.
Find $(g\circ f)(x)$ if $f(x)=x^2-1$ and $g(x)=x+3$.
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$(g\circ f)(x)=x^2+2$. Add 3 to $f(x)=x^2-1$ to get $x^2+2$.
$(g\circ f)(x)=x^2+2$. Add 3 to $f(x)=x^2-1$ to get $x^2+2$.
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Find $(f\circ g)(x)$ if $f(x)=3x$ and $g(x)=\frac{x}{2}+5$.
Find $(f\circ g)(x)$ if $f(x)=3x$ and $g(x)=\frac{x}{2}+5$.
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$(f\circ g)(x)=\frac{3}{2}x+15$. Apply $f(x)=3x$ to $g(x)=\frac{x}{2}+5$.
$(f\circ g)(x)=\frac{3}{2}x+15$. Apply $f(x)=3x$ to $g(x)=\frac{x}{2}+5$.
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Find $(g\circ f)(x)$ if $f(x)=3x$ and $g(x)=\frac{x}{2}+5$.
Find $(g\circ f)(x)$ if $f(x)=3x$ and $g(x)=\frac{x}{2}+5$.
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$(g\circ f)(x)=\frac{3}{2}x+5$. Apply $g$ to $f(x)=3x$ to get $\frac{3x}{2}+5$.
$(g\circ f)(x)=\frac{3}{2}x+5$. Apply $g$ to $f(x)=3x$ to get $\frac{3x}{2}+5$.
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What is $(f\circ g)(-1)$ if $f(x)=x^2$ and $g(x)=x+4$?
What is $(f\circ g)(-1)$ if $f(x)=x^2$ and $g(x)=x+4$?
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$(f\circ g)(-1)=9$. $g(-1)=3$, then $f(3)=9$.
$(f\circ g)(-1)=9$. $g(-1)=3$, then $f(3)=9$.
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What is $(g\circ f)(-1)$ if $f(x)=x^2$ and $g(x)=x+4$?
What is $(g\circ f)(-1)$ if $f(x)=x^2$ and $g(x)=x+4$?
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$(g\circ f)(-1)=5$. $f(-1)=1$, then $g(1)=5$.
$(g\circ f)(-1)=5$. $f(-1)=1$, then $g(1)=5$.
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