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

The function $f$ is given by $f\left(x\right)=2x+1$. Let $f^{-1}$ be the inverse function of $f$. What is the value of $f\left(f^{-1}\left(7\right)\right)$?

The function $f$ has domain $\left[2,8\right]$ and range $\left[-3,5\right]$. Let $f^{-1}$ be the inverse function of $f$. What is the domain of $f^{-1}$?

The function $f$ is given by $f\left(x\right)=x^2-2$, and the function $g$ is given by $g\left(x\right)=\frac{x+1}{x}$. Which of the following is an expression for $f\left(g\left(x\right)\right)$ ?

The table gives values for the functions $f$ and $g$ at selected values of $x$. Functions $f$ and $g$ are defined for all real numbers. Let $h$ be the function defined by $h\left(x\right)=f\left(g\left(x\right)\right)$. What is the value of $h\left(1\right)$ ?

The graph of the piecewise-linear function $f$ is shown in the figure. Let $g$ be the inverse function of $f$. What is the maximum value of $g$ ?

The function $h$ is given by $h\left(x\right)=\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)$ and has a period of $4\pi$. In order to define the inverse function of $h$, which of the following specifies a restricted domain for $h$ and provides a rationale for why $h$ is invertible on that domain?

The table gives values of the function $g$ for selected values of $x$. The function $f$ is given by $f\left(x\right)=2^x+x^2$. What is the value of $f\left(g\right(3\left)\right)$?

The function $f$ is given by $f\left(x\right)=x^2+1$, and the function $g$ is given by $g\left(x\right)=\frac{x-1}{x}$. Which of the following is an expression for $f\left(g\left(x\right)\right)$?

A factory produces widgets. The cost per widget, in dollars, depends on the number of widgets produced, $q$, according to the function $P$ given by $P\left(q\right)=12-0.03q$. The number of widgets produced each day depends on the day's temperature $T$, in degrees Celsius, according to the function $q$ given by $q\left(T\right)=80+1.5T$.

When the temperature is $T=22$ degrees Celsius, what is the cost per widget, in dollars?

The functions $f$ and $g$ are given by $f\left(x\right)=\sqrt{x-1}$ and $g\left(x\right)=x^2$. Which of the following is the domain of the composite function $f\circ g$?

The function $f$ is given by $f\left(x\right)=\frac{2x}{x-1}$ for $x\ne 1$. Which of the following is an expression for $f^{-1}\left(x\right)$?

The function $f$ is given by $f\left(x\right)=x^2-4$ for $x\ge 0$. Which of the following is an expression for $f^{-1}\left(x\right)$?

The functions $f$ and $g$ are given by $f\left(x\right)=\sqrt{2x+5}$ and $g\left(x\right)=3x-1$. What is the value of $\left(f\circ g\right)\left(4\right)$?

The function $f$ is given by $f\left(x\right)=x^3-5$. What is the value of $f^{-1}\left(20\right)$?

Let $h\left(x\right)=\sqrt{x^2+4}$ and $f\left(x\right)=\sqrt{x+4}$. If $h\left(x\right)=\left(f\circ g\right)\left(x\right)$ for all real $x$, which of the following is the function $g$?

The function $f$ is given by $f\left(x\right)=x^2-4x+1$. In order to define the inverse function $f^{-1}$, which of the following specifies a restricted domain for $f$?

The function $f$ is an invertible function such that $f\left(3\right)=7$ and $f^{-1}\left(3\right)=5$. What is the value of $f\left(f\left(5\right)\right)$?

The temperature $T$, in degrees, of a chemical reaction is modeled by the function $h$ given by $h\left(p\right)=2p^2+30$ for $p>0$, where $p$ is the pressure in atmospheres. The pressure in atmospheres is modeled by the function $g$ given by $g\left(t\right)=\frac{t}{2}+1$ for $t\ge 0$, where $t$ is the time in seconds. At what value of $t$ does the temperature reach $70$ degrees?

Let $f$ be the function defined by $f\left(x\right)=\frac{3+2x}{1-x}$. Which of the following is equal to $\left(f\circ f\right)\left(x\right)$?

Let $f$ be the function defined by $f\left(x\right)={\left(x-3\right)}^2+1$ for $x\ge 3$. What is the value of $f^{-1}\left(7\right)$?