Exercises: Injective, surjective and bijective functions and inverses

Find the inverse function $f^{-1}(x)$ of $f(x) = 4x - 7$, where $f:\mathbb{R}\to\mathbb{R}$. Verify that $f(f^{-1}(x)) = x$.

State whether each function $f:\mathbb{R}\to\mathbb{R}$ is injective (one-to-one), giving a reason.

1. a) $f(x) = 3x + 1$
2. b) $f(x) = x^2$
3. c) $f(x) = x^3$

Let $f:\mathbb{R}\to\mathbb{R}$, $f(x) = x^2 + 1$.

1. a) Find the range of $f$.
2. b) Is $f$ surjective? Give a reason.
3. c) Is $f$ bijective? Give a reason.

Let $f(x) = \dfrac{2x+1}{x-3}$ for $x \in \mathbb{R}\setminus\{3\}$.

1. a) Find $f^{-1}(x)$.
2. b) State the domain of $f^{-1}$.
3. c) Find the values of $x$ for which $f(x) = f^{-1}(x)$, i.e. solve $f(x)=f^{-1}(x)$ and describe the result.

Let $f(x) = x^2 - 6x + 5$ with domain $x \geq 3$.

1. a) Write $f(x)$ in the form $(x-p)^2 + q$.
2. b) Explain why $f$ is injective on this domain.
3. c) Find $f^{-1}(x)$ and state its domain.

The temperature in degrees Celsius in a classroom in Cork is converted to Fahrenheit by $F(C) = 1.8C + 32$.

1. a) Find the inverse function $C(F)$.
2. b) Find the Celsius temperature when it is 68°F.
3. c) Show that the graphs of $F$ and its inverse, when drawn on the same axes with the same variable $x$, meet at exactly one point, and find that point.

Let $f:\mathbb{R}\to\mathbb{R}$, $f(x) = x^3 - 3x$.

1. a) Show that $f$ is not injective by finding two different values of $x$ with the same image.
2. b) Find the largest interval of the form $[1, \infty)$ or a subset on which $f$ is injective, justifying using the derivative.
3. c) State a codomain for $f$ restricted to $[1,\infty)$ so that it is bijective.

Let $f(x) = e^{2x} + 1$ for $x \in \mathbb{R}$.

1. a) Prove that $f$ is injective.
2. b) Find the range of $f$ and hence a codomain making $f$ bijective.
3. c) Find $f^{-1}(x)$, state its domain, and solve $f(x) = f^{-1}(x+2)$ or explain why this cannot easily be solved algebraically; instead show that $f^{-1}(f(x)) = x$.

Let $f(x) = \dfrac{ax+b}{cx+d}$ with $ad - bc \neq 0$ and $c\neq 0$.

1. a) Show that $f$ is injective on its domain.
2. b) Find $f^{-1}(x)$.
3. c) Show that $f = f^{-1}$ if and only if $a = -d$.