Chapter 3: Functions

Introduction to Functions

A function $f:D \rightarrow C$ is a rule that assigns to each element $x$ in a set $D$ exactly one element, called $f(x)$, in a set $C$.

• $D$ is called the domain of $f$.
• $C$ is called the codomain of $f$.
• The subset $R=\left\{ y \in C \mid f(x)=y \textrm{ for some }x \in D \right\}$ is called the range of $f$.

Intercepts

Use the following rules to determine the intercepts, or zeros, of a given function $y=f(x)$.

1. To determine the $x$-intercepts, set $y=0$ and solve for $x$.
2. To determine the $y$-intercepts, set $x=0$ and solve for $y$.

Summary of Important Functions

The following is a summary of important functions. You should have this information memorized for your calculus class. Links are given to the sections of the workshop where more information can be found.

1. Basic Linear Function: $f(x) = x$ (more on linear functions)
   • Domain: $(-\infty,\infty) = \left\{ x \mid x \in \mathbb{R}\right\}$
   • Range: $(-\infty,\infty) = \left\{ y \mid y \in \mathbb{R}\right\}$

2. Basic Quadratic Function: $f(x) = x^{2}$ (more on quadratic functions)
   • Domain: $(-\infty,\infty) = \left\{ x \mid x \in \mathbb{R}\right\}$
   • Range: $[0, \infty) = \left\{ y \mid y \geq 0 \right\}$

3. Basic Cubic Function: $f(x) = x^{3}$ (more on polynomial functions)
   • Domain: $(-\infty,\infty) = \left\{ x \mid x \in \mathbb{R}\right\}$
   • Range: $(-\infty,\infty) = \left\{ y \mid y \in \mathbb{R}\right\}$

4. Basic Square Root Function: $f(x) = \sqrt{x}$ (more on radical functions)
   • Domain: $[0, \infty) = \left\{ x \mid x \geq 0 \right\}$
   • Range: $[0, \infty) = \left\{ y \mid y \geq 0 \right\}$

5. Basic Cube Root Function: $f(x) = \sqrt[3]{x}$
   • Domain: $(-\infty,\infty) = \left\{ x \mid x \in \mathbb{R}\right\}$
   • Range: $(-\infty,\infty) = \left\{ y \mid y \in \mathbb{R}\right\}$

6. Reciprocal Function: $\displaystyle f(x) =\frac{1}{x}$ (more on rational functions)
   • Domain: $(-\infty,0) \cup (0,\infty) = \left\{ x \mid x \neq 0 \right\}$
   • Range: $(-\infty,0) \cup (0,\infty) = \left\{ y \mid y \neq 0 \right\}$

7. Absolute Value Function: $\displaystyle f(x) = |x| = \begin{cases} x & \text{ if } x\geq 0 \\ -x & \text{ if } x < 0 \end{cases}$
   (more on absolute value functions)
   • Domain: $(-\infty,\infty) = \left\{ x \mid x \in \mathbb{R}\right\}$
   • Range: $[0, \infty) = \left\{ y \mid y \geq 0 \right\}$

8. Exponential Function with Base $a>0$ and $a\neq1$: $f(x) = a^{x}$ (more on exponential functions)
   • Domain: $(-\infty,\infty) = \left\{ x \mid x \in \mathbb{R}\right\}$
   • Range: $(0, \infty) = \left\{ y \mid y > 0 \right\}$

9. Logarithmic Function with Base $a>0$ and $a\neq1$: $f(x) = \log_{a}(x)$ (more on logarithmic functions)
   • Domain: $(0,\infty) = \left\{ x \mid x >0 \right\}$
   • Range: $(-\infty, \infty) = \left\{ y \mid y \in \mathbb{R} \right\}$

10. Sine Function: $f(x) = \sin(x)$ (more on trigonometric functions)
    • Domain: $(-\infty,\infty) = \left\{ x \mid x \in \mathbb{R}\right\}$
    • Range: $[-1,1] = \left\{ y \mid -1\leq y \leq 1 \right\}$

11. Cosine Function: $f(x) = \cos(x)$
    • Domain: $(-\infty,\infty) = \left\{ x \mid x \in \mathbb{R}\right\}$
    • Range: $[-1,1] = \left\{ y \mid -1\leq y \leq 1 \right\}$

12. Tangent Function: $f(x) = \tan(x)$
    • Domain: $\cdots \cup \left( -\frac{3\pi}{2},-\frac{\pi}{2} \right) \cup \left( -\frac{\pi}{2},\frac{\pi}{2} \right) \cup \left( \frac{\pi}{2},\frac{3\pi}{2} \right) \cup \cdots = \left\{ x \mid x \neq \frac{\pi}{2} + k\pi, k \in \mathbb{Z}\right\}$
    • Range: $(-\infty, \infty) = \left\{ y \mid y \in \mathbb{R} \right\}$