### Key Concepts

## 1.1 Review of Functions

- A function is a mapping from a set of inputs to a set of outputs with exactly one output for each input.
- If no domain is stated for a function $y=f\left(x\right),$ the domain is considered to be the set of all real numbers $x$ for which the function is defined.
- When sketching the graph of a function $f,$ each vertical line may intersect the graph, at most, once.
- A function may have any number of zeros, but it has, at most, one
*y*-intercept. - To define the composition $g\circ f,$ the range of $f$ must be contained in the domain of $g.$
- Even functions are symmetric about the $y$-axis whereas odd functions are symmetric about the origin.

## 1.2 Basic Classes of Functions

- The power function $f\left(x\right)={x}^{n}$ is an even function if $n$ is even and $n\ne 0,$ and it is an odd function if $n$ is odd.
- The root function $f\left(x\right)={x}^{1\text{/}n}$ has the domain $[0,\infty )$ if $n$ is even and the domain $(\text{\u2212\u221e},\infty )$ if $n$ is odd. If $n$ is odd, then $f\left(x\right)={x}^{1\text{/}n}$ is an odd function.
- The domain of the rational function $f\left(x\right)=p\left(x\right)\text{/}q\left(x\right),$ where $p\left(x\right)$ and $q\left(x\right)$ are polynomial functions, is the set of $x$ such that $q\left(x\right)\ne 0.$
- Functions that involve the basic operations of addition, subtraction, multiplication, division, and powers are algebraic functions. All other functions are transcendental. Trigonometric, exponential, and logarithmic functions are examples of transcendental functions.
- A polynomial function $f$ with degree $n\ge 1$ satisfies $f\left(x\right)\to \text{\xb1}\infty $ as $x\to \text{\xb1}\infty .$ The sign of the output as $x\to \infty $ depends on the sign of the leading coefficient only and on whether $n$ is even or odd.
- Vertical and horizontal shifts, vertical and horizontal scalings, and reflections about the $x$- and $y$-axes are examples of transformations of functions.

## 1.3 Trigonometric Functions

- Radian measure is defined such that the angle associated with the arc of length 1 on the unit circle has radian measure 1. An angle with a degree measure of $180\text{\xb0}$ has a radian measure of $\pi $ rad.
- For acute angles $\theta ,$ the values of the trigonometric functions are defined as ratios of two sides of a right triangle in which one of the acute angles is $\theta .$
- For a general angle $\theta ,$ let $(x,y)$ be a point on a circle of radius $r$ corresponding to this angle $\theta .$ The trigonometric functions can be written as ratios involving $x,y,$ and $r.$
- The trigonometric functions are periodic. The sine, cosine, secant, and cosecant functions have period $2\pi .$ The tangent and cotangent functions have period $\pi .$

## 1.4 Inverse Functions

- For a function to have an inverse, the function must be one-to-one. Given the graph of a function, we can determine whether the function is one-to-one by using the horizontal line test.
- If a function is not one-to-one, we can restrict the domain to a smaller domain where the function is one-to-one and then define the inverse of the function on the smaller domain.
- For a function $f$ and its inverse ${f}^{\mathrm{-1}},f\left({f}^{\mathrm{-1}}\left(x\right)\right)=x$ for all $x$ in the domain of ${f}^{\mathrm{-1}}$ and ${f}^{\mathrm{-1}}\left(f\left(x\right)\right)=x$ for all $x$ in the domain of $f.$
- Since the trigonometric functions are periodic, we need to restrict their domains to define the inverse trigonometric functions.
- The graph of a function $f$ and its inverse ${f}^{\mathrm{-1}}$ are symmetric about the line $y=x.$

## 1.5 Exponential and Logarithmic Functions

- The exponential function $y={b}^{x}$ is increasing if $b>1$ and decreasing if $0<b<1.$ Its domain is $(\text{\u2212}\infty ,\infty )$ and its range is $\left(0,\infty \right).$
- The logarithmic function $y={\text{log}}_{b}(x)$ is the inverse of $y={b}^{x}.$ Its domain is $\left(0,\infty \right)$ and its range is $\left(\text{\u2212}\infty ,\infty \right).$
- The natural exponential function is $y={e}^{x}$ and the natural logarithmic function is $y=\text{ln}\phantom{\rule{0.1em}{0ex}}x={\text{log}}_{e}x.$
- Given an exponential function or logarithmic function in base $a,$ we can make a change of base to convert this function to any base $b>0,b\ne 1.$ We typically convert to base $e.$
- The hyperbolic functions involve combinations of the exponential functions ${e}^{x}$ and ${e}^{\text{\u2212}x}.$ As a result, the inverse hyperbolic functions involve the natural logarithm.