Composite Functions: Questions with Detailed Solutions

This page presents a carefully selected set of questions on the composition of functions, each followed by a clear and detailed solution. The goal is to strengthen conceptual understanding and improve computational skills.

Question 1

Let $f(x) = 2x + 3$ and $g(x) = -x^2 + 1$. Find the composite function $(f \circ g)(x)$.

Solution

By definition, \[ (f \circ g)(x) = f(g(x)). \] Substitute $g(x)$ into $f$: \[ (f \circ g)(x) = 2(-x^2 + 1) + 3 = -2x^2 + 5. \]

Question 2

Given $f(2)=3$, $g(3)=2$, $f(3)=4$, and $g(2)=5$, evaluate $(f \circ g)(3)$.

Solution

\[ (f \circ g)(3) = f(g(3)) = f(2) = 3. \]

Question 3

Let \[ f(x) = \sqrt{x+2}, \quad g(x) = \ln(1 - x^2). \] Find $(g \circ f)(x)$ and determine its domain.

Solution

\[ (g \circ f)(x) = g(f(x)) = \ln\!\bigl(1 - (\sqrt{x+2})^2\bigr) = \ln(-x - 1). \]

Domain:

The domain is the intersection of the above sets: \[ [-2,-1). \]

Question 4

Let \[ f = \{(-2,1),(0,3),(4,5)\}, \quad g = \{(1,1),(3,3),(7,9)\}. \] Find $g \circ f$, and state its domain and range.

Solution

\[ (g \circ f)(-2) = g(1) = 1, \quad (g \circ f)(0) = g(3) = 3. \] \[ (g \circ f)(4) = g(5) \text{ is undefined.} \]

\[ g \circ f = \{(-2,1),(0,3)\} \]

Domain: $\{-2,0\}$ Range: $\{1,3\}$

Question 5

Let $f(x)=\ln x$. Find the derivative of \[ F(x) = (f \circ f)(x). \]

Solution

\[ F(x) = \ln(\ln x). \] Using the chain rule: \[ F'(x) = \frac{1}{x\ln x}. \]

Question 6

Write \[ F(x) = |4x^2 + 2x - 5| \] as the composition of two functions.

Solution

One possible choice: \[ g(x) = 4x^2 + 2x - 5, \quad f(x) = |x|. \] Then: \[ F(x) = f(g(x)). \]

Question 7

Given $g(x)=\dfrac{1}{x}$ and \[ F(x) = \frac{1/x}{1+x}, \] write $F$ as a composite function.

Solution

Let: \[ f(x) = \frac{x}{1 + 1/x}. \] Then: \[ F(x) = f(g(x)). \]

Question 8

Let \[ f(x)= \begin{cases} x, & x<0 \\ x^2, & x\ge 0 \end{cases} \quad\text{and}\quad g(x)=\sqrt{x}. \] Find $g(f(x))$.

Solution

\[ g(f(x)) = x \quad \text{for } x \ge 0. \]

Question 9

True or False: $f(g(x)) = g(f(x))$ for all functions $f$ and $g$.

Solution

False. Example: \[ f(x)=x+1,\quad g(x)=x^2. \] \[ f(g(x))=x^2+1,\quad g(f(x))=(x+1)^2. \]

Question 10

Evaluate $f(g(h(1)))$ if \[ h(x)=-|x|,\quad g(x)=x-1,\quad f(x)=\frac{1}{x+2}. \]

Solution

\[ h(1)=-1,\quad g(-1)=-2. \] \[ f(-2) \text{ is undefined.} \]

Therefore, $f(g(h(1)))$ is undefined.

Exercises

  1. Evaluate $f(g(3))$ if $f(x)=|x-6|+x^2-1$ and $g(x)=2x$.
  2. Find $f$ and $g$ if $f(g(x))=2\sec(2x+1)$.
  3. Find the domain of $g\circ f$ if $f(x)=\sqrt{x}$ and $g(x)=1/x$.
  4. Find the range of $f(g(x))$ if $f(x)=x+4$ and $g(x)=x^2+2$.

Answers

  1. $35$
  2. $f(x)=2\sec x,\; g(x)=2x+1$
  3. $(0,\infty)$
  4. $[6,\infty)$

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