Compose Two Functions - Algebra 1

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

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

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

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

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

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

4. State the associative property for composition using $f$, $g$, and $h$.

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

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

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

6. What does the notation $(f\circ g)(x)$ mean in words?

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

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

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

8. What is $(g\circ f)(2)$ if $f(x)=x-5$ and $g(x)=3x$?

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

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

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

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

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

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

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

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

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

13. Identify the inner function in $f(g(x))$.

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

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

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

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

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

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

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

17. What is $(g\circ f)(x)$ if $f(x)=x^3$ and $g(x)=2x$?

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

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

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

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

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

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

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

21. What is the meaning of $T(h(t))$ if $T$ depends on height and $h$ depends on time?

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

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

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

23. What is $(f\circ g)(x)$ if $f(x)=x^3$ and $g(x)=2x$?

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

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

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

25. What is the domain of $(f\circ g)$ described as a set condition?

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

26. What is the key difference between $(f\circ g)(x)$ and $(g\circ f)(x)$?

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

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

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

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

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

29. What is the definition of the composition $(g\circ f)(x)$?

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

30. What is the definition of the composition $(f\circ g)(x)$ in terms of $f$ and $g$?

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

31. What does $(f\circ g)(a)$ mean when $a$ is a number?

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

32. What is the identity function $I(x)$ used for 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

48. What is a common notation for composing $f$ with $g$ besides $(f\circ g)(x)$?

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

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

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

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

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

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

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

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