This page presents a carefully selected set of questions on the composition of functions. Each solution is hidden in a collapsible dropdown so you can practice independently before reviewing the steps[cite: 1].
Let \(f(x) = 2x + 3\) and \(g(x) = -x^2 + 1\). Find the composite function \((f \circ g)(x)\).
By definition, \[ (f \circ g)(x) = f(g(x)). \] Substitute \(g(x)\) into \(f\): \[ (f \circ g)(x) = 2(-x^2 + 1) + 3 = -2x^2 + 5. \]
Given \(f(2)=3\), \(g(3)=2\), \(f(3)=4\), and \(g(2)=5\), evaluate \((f \circ g)(3)\).
Let \[ f(x) = \sqrt{x+2}, \quad g(x) = \ln(1 - x^2). \] Find \((g \circ f)(x)\) and determine its domain.
Domain:
The domain is the intersection of the above sets: \[ [-2,-1). \]
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.
\[ g \circ f = \{(-2,1),(0,3)\} \]
Domain: \(\{-2,0\}\)
Range: \(\{1,3\}\)
Let \(f(x)=\ln x\). Find the derivative of \[ F(x) = (f \circ f)(x). \]
Write \[ F(x) = |4x^2 + 2x - 5| \] as the composition of two functions.
One possible choice:
\[ g(x) = 4x^2 + 2x - 5, \quad f(x) = |x|. \]Then:
\[ F(x) = f(g(x)). \]Given \(g(x)=\dfrac{1}{x}\) and \[ F(x) = \frac{1/x}{1+x}, \] write \(F\) as a composite function.
Let:
\[ f(x) = \frac{x}{1 + 1/x}. \]Then:
\[ F(x) = f(g(x)). \]Let \[ f(x)= \begin{cases} x, & x<0 \\ x^2, & x\ge 0 \end{cases} \quad\text{and}\quad g(x)=\sqrt{x}. \] Find \(g(f(x))\).
True or False: \(f(g(x)) = g(f(x))\) for all functions \(f\) and \(g\).
False.
Example:
Evaluate \(f(g(h(1)))\) if \[ h(x)=-|x|,\quad g(x)=x-1,\quad f(x)=\frac{1}{x+2}. \]
Therefore, \(f(g(h(1)))\) is undefined.