Exercises: Types of functions, inverses and composites

Let $f:\mathbb{R}\to\mathbb{R}$ be given by $f(x) = 3x - 5$ and $g:\mathbb{R}\to\mathbb{R}$ by $g(x) = x^2 + 1$.

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

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

1. c) Find expressions for $f(g(x))$ and $g(f(x))$.

Find the inverse function $f^{-1}(x)$ of $f(x) = \frac{2x+1}{3}$, $x \in \mathbb{R}$. Verify that $f(f^{-1}(x)) = x$.

Write $f(x) = x^2 - 6x + 11$ in the form $(x-a)^2 + b$ by completing the square. Hence state the minimum value of $f$ and the value of $x$ at which it occurs.

The function $f:\mathbb{R}\to\mathbb{R}$ is defined by $f(x) = x^2 - 4x + 7$.

1. a) Is $f$ injective? Justify your answer.

1. b) Is $f$ surjective? Justify your answer.

1. c) Restrict the domain to $[2, \infty)$ and the codomain to $[3,\infty)$. Show that $f$ is then bijective and find $f^{-1}(x)$.

Let $f(x) = e^{2x}$ for $x\in\mathbb{R}$.

1. a) State the range of $f$.

1. b) Find $f^{-1}(x)$ and state its domain.

1. c) Solve $f(x) = 5$, giving your answer correct to three decimal places.

The cost in euro of hiring a boat on Lough Derg for $h$ hours is $C(h) = 15h + 20$. A discount function $D(c) = 0.9c$ applies to any cost $c$.

1. a) Find $D(C(h))$ and interpret it.

1. b) Find the inverse of $D\circ C$ and use it to find how many hours can be hired for €72.

Let $f(x) = \frac{x+2}{x-1}$ for $x\in\mathbb{R}$, $x\neq 1$.

1. a) Show that $f$ is injective.

1. b) Find $f^{-1}(x)$ and show that $f^{-1} = f$.

1. c) Hence find $f(f(x))$ and explain the result.

Let $f(x) = 2x^2 + 8x + 3$ for $x \geq -2$.

1. a) Express $f(x)$ in the form $a(x+p)^2 + q$.

1. b) Find the range of $f$ and show that $f$ is bijective onto its range.

1. c) Find $f^{-1}(x)$ and sketch, in words, the relationship between the graphs of $f$ and $f^{-1}$. Find the point where the graph of $f^{-1}$ meets the line $y = x$, if such a point exists.

Let $f(x) = \ln(x-1)$ for $x>1$ and $g(x) = e^x + 1$ for $x \in\mathbb{R}$.

1. a) Show that $f$ and $g$ are inverses of each other.

1. b) Find $f(g(x))$ and $g(f(x))$, stating the domain in each case.

1. c) Solve $f(x) + f(x+1) = \ln 6$ for $x$.