1.2 Basic classes of functions  (Page 10/28)

We can summarize the different transformations and their related effects on the graph of a function in the following table.

Key concepts

• 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 $(\text{−∞},\infty )$ 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 $f\left(x\right)\to \text{±}\infty$ as $x\to \text{±}\infty .$ 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.

Key equations

• Point-slope equation of a line
  $y-y_1=m\left(x-x_1\right)$
• Slope-intercept form of a line
  $y=mx+b$
• Standard form of a line
  $ax+by=c$
• Polynomial function
  $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\text{⋯}+a_{1}x+a_0$

For the following exercises, for each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical.

For the following exercises, write the equation of the line satisfying the given conditions in slope-intercept form.