Composite & Inverse Functions (College Board AP® Precalculus): Free Response Questions

The figure shows the graph of the function $f$ on its domain of $-3.5\le x\le 3.5$. The points $(-3,1)$, $(0,1)$, and $(3,1)$ are on the graph of $f$. The function $g$ is given by $g\left(x\right)=2.916·{\left(0.7\right)}^x$.

(i) The function $h$ is defined by $h\left(x\right)=\left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)$. Find the value of $h\left(3\right)$ as a decimal approximation, or indicate that it is not defined.

(ii) Find all values of $x$ for which $f\left(x\right)=1$, or indicate that there are no such values.

(i) Determine if $f$ has an inverse function.

(ii) Give a reason for your answer based on the definition of a function and the graph of $y=f\left(x\right)$.

The function $h$ is given by $h\left(x\right)={\left(2x-5\right)}^3$.

Express $h\left(x\right)$ as a composition $f\left(g\left(x\right)\right)$, where $f$ and $g$ are functions and neither $f$ nor $g$ is the identity function. Give explicit expressions for $f\left(x\right)$ and $g\left(x\right)$.

The function $f$ is given by $f\left(x\right)=x^3+2$. Let $f^{-1}$ be the inverse function of $f$.

Find the exact value of $f\left(f^{-1}\left(11\right)\right)$.

The function $f$ is increasing and is defined for $0\le x\le 5$. The range of $f$ is $-2\le y\le 18$. Let $f^{-1}$ be the inverse function of $f$.

State the domain and range of $f^{-1}$.

The graph of a function $f$ in the $xy$-plane passes through the points $\left(-1,0\right)$ and $\left(2,5\right)$. Let $f^{-1}$ be the inverse function of $f$.

State the coordinates of two points that must lie on the graph of $f^{-1}$. Then describe how the graph of $f^{-1}$ relates geometrically to the graph of $f$.

The function $f$ is increasing and is defined for $x>0$. The table gives values of $f\left(x\right)$ at selected values of $x$.

The function $g$ is given by $g\left(x\right)=-0.1371x^4+x^3-1.2x^2$.

(i) The function $h$ is defined by $h\left(x\right)=\left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)$. Find the value of $h\left(2\right)$ as a decimal approximation, or indicate that it is not defined. Show the work that leads to your answer.

(ii) Find the value of $f^{-1}\left(7\right)$, or indicate that it is not defined.

The function $f$ is increasing and is defined for $x\ge 0$. The table gives values of $f\left(x\right)$ at selected values of $x$.

The function $g$ is given by $g\left(x\right)=\frac{2x^3-5x+1}{x-3}$.

(i) The function $h$ is defined by $h\left(x\right)=\left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)$. Find the value of $h\left(2\right)$ as a decimal approximation, or indicate that it is not defined. Show the work that leads to your answer.

(ii) Find the value of $f^{-1}\left(5\right)$, or indicate that it is not defined.

The function $f$ is given by $f\left(x\right)=\frac{3x+1}{x-2}$ for $x\ne 2$.

Find an expression for $f^{-1}\left(x\right)$. Show the work that leads to your answer.

The number of mugs a small company sells per week depends on the amount spent on advertising, $x$ dollars, according to the function $N$ given by $N\left(x\right)=30+0.05x$.

The weekly revenue, in dollars, depends on the number of mugs sold, $N$, according to the function $R$ given by $R\left(N\right)=7.50·N-80$.

Find the value of $\left(R\circ N\right)\left(400\right)$. Interpret the meaning of your answer in the context of the problem.

The function $f$ is given by $f\left(x\right)=\sqrt{x-5}$ for $x\ge 5$. The function $g$ is given by $g\left(x\right)=x^2+1$ for all real numbers $x$.

(i) Find a simplified expression for $\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$.

(ii) State the domain of $f\circ g$.

The function $f$ is given by $f\left(x\right)=\frac{2x+3}{x-1}$ for $x\ne 1$. The function $g$ is given by $g\left(x\right)=x^2+4$ for $x\ge 0$.

(i) Find an expression for $f^{-1}\left(x\right)$.

(ii) Find the value of $f^{-1}\left(g\left(2\right)\right)$ as a decimal approximation. Show the work that leads to your answer.

The function $f$ is an increasing function defined for all real numbers. The table gives values of $f$ at selected values of $x$.

The function $g$ is given by $g\left(x\right)=x^2-2x+5$ for $x\ge 1$.

(i) Find the value of $f^{-1}\left(2\right)$.

(ii) Find the value of $g^{-1}\left(f\left(0\right)\right)$ as a decimal approximation. Show the work that leads to your answer.