Composition of Functions - AP Precalculus

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

   $(g \bigcirc f)(x) = 3(x + 2)$. Apply $g$ to $f(x) = x + 2$: $3(x + 2)$.

2. What is the definition of the composition of functions?

   It is applying one function to the results of another: $(f \bigcirc g)(x) = f(g(x))$. First apply g, then apply f to that result.

3. If $f(x) = x - 3$ and $g(x) = x^2$, find $(f \bigcirc g)(x)$.

   $(f \bigcirc g)(x) = x^2 - 3$. Apply $f$ to $g(x) = x^2$: $x^2 - 3$.

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

   $(f \bigcirc g)(x) = 9x^2 + 1$. Substitute $g(x) = 3x$ into $f$: $(3x)^2 + 1$.

5. What is $(f \bigcirc g)(0)$ for $f(x) = 2x + 3$ and $g(x) = x^2$?

   $(f \bigcirc g)(0) = 3$. $g(0) = 0$, then $f(0) = 3$.

6. If $f(x) = x^2$ and $g(x) = 3x + 1$, find $(g \bigcirc f)(x)$.

   $(g \bigcirc f)(x) = 3x^2 + 1$. Apply $g$ to $f(x) = x^2$: $g(x^2) = 3x^2 + 1$.

7. What is $(g \bigcirc f)(x)$ for $f(x) = \frac{1}{x}$ and $g(x) = x^2$?

   $(g \bigcirc f)(x) = \frac{1}{x^2}$. Apply $g$ to $f(x) = \frac{1}{x}$: $(\frac{1}{x})^2$.

8. If $f(x) = 4x$ and $g(x) = x - 2$, find $(f \bigcirc g)(x)$.

   $(f \bigcirc g)(x) = 4(x - 2)$. Apply $f$ to $g(x) = x - 2$: $4(x - 2)$.

9. Write $(f \bigcirc g)(x)$ for $f(x) = x^3$ and $g(x) = \frac{1}{x}$.

   $(f \bigcirc g)(x) = (1/x)^3$. Substitute $g(x) = \frac{1}{x}$ into $f(x) = x^3$.

10. If $f(x) = x^2$ and $g(x) = x + 1$, what is $(f \bigcirc g)(1)$?

    $(f \bigcirc g)(1) = 4$. $g(1) = 2$, then $f(2) = 4$.

11. Determine $(f \bigcirc g)(x)$ for $f(x) = x^2$ and $g(x) = x + 1$.

    $(f \bigcirc g)(x) = (x + 1)^2$. Apply $f$ to $g(x) = x + 1$: $(x + 1)^2$.

12. If $f(x) = 5x$ and $g(x) = x - 3$, find $(f \bigcirc g)(x)$.

    $(f \bigcirc g)(x) = 5(x - 3)$. Substitute $g(x) = x - 3$ into $f(x) = 5x$.

13. Determine $(f \bigcirc g)(1)$ if $f(x) = x - 2$ and $g(x) = x^2$.

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

14. What is $(f \bigcirc g)(x)$ for $f(x) = x^3$ and $g(x) = \frac{1}{x}$?

    $(f \bigcirc g)(x) = \frac{1}{x^3}$. Apply $f$ to $g(x) = \frac{1}{x}$: $(\frac{1}{x})^3$.

15. If $f(x) = \frac{1}{x}$ and $g(x) = x + 2$, find $(f \bigcirc g)(x)$.

    $(f \bigcirc g)(x) = \frac{1}{x + 2}$. Apply $f$ to $g(x) = x + 2$: $\frac{1}{x + 2}$.

16. What is $(g \bigcirc f)(5)$ for $f(x) = 3x$ and $g(x) = x - 2$?

    $(g \bigcirc f)(5) = 13$. $f(5) = 15$, then $g(15) = 13$.

17. If $f(x) = x^2$ and $g(x) = x + 2$, find $(g \bigcirc f)(x)$.

    $(g \bigcirc f)(x) = x^2 + 2$. Apply $g$ to $f(x) = x^2$: $x^2 + 2$.

18. Find $(f \bigcirc g)(-1)$ if $f(x) = x^2$ and $g(x) = x + 5$.

    $(f \bigcirc g)(-1) = 16$. $g(-1) = 4$, then $f(4) = 16$.

19. If $f(x) = x^2 + 2$ and $g(x) = 4x$, find $(f \bigcirc g)(x)$.

    $(f \bigcirc g)(x) = 16x^2 + 2$. Substitute $g(x) = 4x$ into $f$: $(4x)^2 + 2$.

20. Calculate $(g \bigcirc f)(3)$ for $f(x) = 2x$ and $g(x) = x^2$.

    $(g \bigcirc f)(3) = 36$. $f(3) = 6$, then $g(6) = 36$.

21. What is $(f \bigcirc g)(x)$ for $f(x) = \frac{1}{2}x$ and $g(x) = x + 4$?

    $(f \bigcirc g)(x) = \frac{1}{2}(x + 4)$. Apply $f$ to $g(x) = x + 4$.

22. Calculate $(g \bigcirc f)(-3)$ for $f(x) = 3x + 1$ and $g(x) = x^2$.

    $(g \bigcirc f)(-3) = 64$. $f(-3) = -8$, then $g(-8) = 64$.

23. Find $(g \bigcirc f)(0)$ for $f(x) = x^2 + 1$ and $g(x) = 2x$.

    $(g \bigcirc f)(0) = 2$. $f(0) = 1$, then $g(1) = 2$.

24. Determine $(f \bigcirc g)(x)$ if $f(x) = 1/x$ and $g(x) = x - 5$.

    $(f \bigcirc g)(x) = 1/(x - 5)$. Apply $f$ to the result $g(x) = x - 5$.

25. State the notation for composing functions $f(x)$ and $g(x)$.

    $(f \bigcirc g)(x) = f(g(x))$. Standard notation for function composition.

26. If $f(x) = 2x + 3$, find $(f \bigcirc f)(x)$.

    $(f \bigcirc f)(x) = 2(2x + 3) + 3$. Substitute $f(x)$ into itself.

27. If $f(x) = x - 4$ and $g(x) = x^2$, find $(f \bigcirc g)(x)$.

    $(f \bigcirc g)(x) = x^2 - 4$. Apply $f$ to $g(x) = x^2$: $x^2 - 4$.

28. Evaluate $(g \bigcirc f)(2)$ if $f(x) = x + 1$ and $g(x) = 2x$.

    $(g \bigcirc f)(2) = 6$. $f(2) = 3$, then $g(3) = 6$.

29. If $f(x) = 2x$ and $g(x) = x + 3$, what is $(f \bigcirc g)(x)$?

    $(f \bigcirc g)(x) = 2(x + 3)$. Substitute $g(x) = x + 3$ into $f$.

30. Find $(g \bigcirc f)(x)$ if $f(x) = x^2$ and $g(x) = \frac{1}{x}$.

    $(g \bigcirc f)(x) = \frac{1}{x^2}$. Apply $g$ to $f(x) = x^2$: $\frac{1}{x^2}$.

31. If $f(x) = 2x + 1$ and $g(x) = x - 3$, find $(f \bigcirc g)(x)$.

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

32. If $f(x) = x + 1$ and $g(x) = x^2$, what is $(f \bigcirc g)(-2)$?

    $(f \bigcirc g)(-2) = 5$. $g(-2) = 4$, then $f(4) = 5$.

33. What is $(f \bigcirc g)(2)$ if $f(x) = 3x$ and $g(x) = x - 1$?

    $(f \bigcirc g)(2) = 3$. $g(2) = 1$, then $f(1) = 3$.

34. Find $(f \bigcirc g)(x)$ if $f(x) = x + 5$ and $g(x) = x^3$.

    $(f \bigcirc g)(x) = x^3 + 5$. Apply $f$ to $g(x) = x^3$: $x^3 + 5$.

35. Calculate $(f \bigcirc g)(x)$ if $f(x) = 2x + 1$ and $g(x) = x^3$.

    $(f \bigcirc g)(x) = 2x^3 + 1$. Apply $f$ to $g(x) = x^3$: $2x^3 + 1$.

36. Determine $(g \bigcirc f)(x)$ if $f(x) = x + 1$ and $g(x) = x^3$.

    $(g \bigcirc f)(x) = (x + 1)^3$. Apply $g$ to $f(x) = x + 1$: $(x + 1)^3$.

Study these flashcards