Exercises: Composite functions

Let $f(x) = 3x + 2$ and $g(x) = x^2 - 1$.

1. a) Find $f(g(2))$.
2. b) Find $g(f(2))$.

Let $f(x) = 2x - 5$ and $g(x) = x + 4$. Find expressions for $(f\circ g)(x)$ and $(g\circ f)(x)$, giving each in the form $ax + b$.

Let $f(x) = x^2$ and $g(x) = x + 3$. Find $f(g(x))$ and $g(f(x))$, and write each in expanded form.

A shop in Galway offers a discount of 20% on all items, and a customer also has a voucher worth €10 that is subtracted from the bill. Let $d(x) = 0.8x$ and $v(x) = x - 10$, where $x$ is the original price in euro.

1. a) Write down $v(d(x))$ and $d(v(x))$.
2. b) For an item originally priced at €60, find the final price under each order of applying the offers.
3. c) Which order is better for the customer? Justify your answer.

Let $f(x) = \dfrac{1}{x-2}$ and $g(x) = x^2 + 1$.

1. a) Find $f(g(x))$ and state its domain.
2. b) Find $g(f(x))$ and state the values of $x$ for which it is defined.
3. c) Solve $f(g(x)) = \dfrac{1}{2}$.

Let $f(x) = 2x + k$ and $g(x) = x^2 + 2$, where $k$ is a constant. Given that $f(g(1)) = g(f(1))$, find the possible values of $k$.

Let $f(x) = ax + b$ where $a, b$ are real constants with $a \neq 0$.

1. a) Show that $f(f(x)) = x$ for all $x$ if and only if either $a = 1$ and $b = 0$, or $a = -1$.
2. b) Hence find $f$ if $f(f(x)) = x$ and $f(3) = 7$.
3. c) Find all linear functions $f$ for which $f(f(x)) = 9x + 8$.

Let $f(x) = \dfrac{x+1}{x-1}$, for $x \neq 1$.

1. a) Show that $f(f(x)) = x$ for all $x \neq 1$.
2. b) Hence write down $f^{-1}(x)$.
3. c) Let $g(x) = 2x - 3$. Solve $(f\circ g)(x) = 3$ and determine $(g\circ f)(x)$ in the form $\dfrac{ax+b}{x-1}$.

Let $f(x) = x^2 - 4x + 7$ for $x \in \mathbb{R}$ and $g(x) = \sqrt{x}$ for $x \geq 0$.

1. a) Find the range of $f$.
2. b) Show that $g(f(x))$ is defined for all real $x$ and find its minimum value.
3. c) Find the range of $f(g(x))$ for $x \geq 0$.