Functions and Their Graphs

Graph each equation.

$y=\sqrt{x}$

Show that a constant function $f\left(x\right)=b$ has an average rate of change of $0$. Evaluate the average rate of change of $y=\sqrt{4-x^2}$ on the interval $[-2,2]$. Explain how this can happen.

The inequality$-1<x<3$ can be written in interval notation  as___________.

If $x=-2$, the value of the expression $3x^2-5x+\frac{1}{x}$ is _______.

The domain of the variable in the expression $\frac{x-3}{x+4}$ is _________.

Solve the inequality: $3-2x>5$. Graph the solution set.

If $f$ is a function defined by the equation $y=f\left(x\right)$, then $x$ is called the ____________ variable and y is the ____________ variable.

The set of all images of the elements in the domain of a function is called the _________.

If the domain of $f$ is all real numbers in the interval $[0,7]$ and the domain of g is all real numbers in the interval $[-2,5]$, the domain of $f+g$ is all real numbers in the interval _________.

If $f\left(x\right)=x+1$and $g\left(x\right)=x^3$, then____________ =$x^3-(x+1)$.

True or False Every relation is a function.

True or False The domain of $(f.g)\left(x\right)$ consists of the numbers $x$ that are in the domains of both $f \mathrm{and} g$.

True or False The independent variable is sometimes

referred to as the argument of the function.

True or False If no domain is specified for a function $f$, then the domain of $f$ is taken to be the set of real numbers.

True or False The domain of the function $f\left(x\right)=\frac{x^2-4}{x}$ is$\left\{x\right|x\ne \pm 2\}$.

Determine whether each relation represents a function. For each function, state the domain and range.

Determine whether each relation represents a function. For each function, state the domain and range.

Determine whether each relation represents a function. For each function, state the domain and range. 

$\left\{\right(2,6), (-3,6), (4,9), (2,10\left)\right\}$

The domain of $\frac{f}{g}$ consists of numbers $x$ for which $g\left(x\right)$ ____ $0$ that are in the domains of both ____ and ____.

In Problems 47– 62, find the domain of each function. 

$h\left(x\right)=\sqrt{3x - 12}$

Determine whether each relation represents a function. For each function, state the domain and range.

Determine whether each relation represents a function. For each function, state the domain and range.

Determine whether each relation represents a function. For each function, state the domain and range. 

$\left\{\right(-2,5), (-1,3), (3,7), (4,12\left)\right\}$

Determine whether each relation represents a function. For each function, state the domain and range. 

$\left\{\right(1,3), (2,3), (3,3), (4,3\left)\right\}$

Determine whether each relation represents a function. For each function, state the domain and range. 

$\left\{\right(0,-2), (1,3), (2,3), (3,7\left)\right\}$

Determine whether each relation represents a function. For each function, state the domain and range. 

$\left\{\right(-2,4), (-2,6), (0,3), (3,7\left)\right\}$

Determine whether each relation represents a function. For each function, state the domain and range. 

$\left\{\right(-4,4), (-3,3), (-2,2), (-1,1), (-4,0\left)\right\}$

Determine whether each relation represents a function. For each function, state the domain and range. 

$\left\{\right(-2,4), (-1,1), (0,0), (1,1\left)\right\}$

Determine whether each relation represents a function. For each function, state the domain and range. 

$\left\{\right(-2,16), (-1,4), (0,3), (1,4\left)\right\}$

Determine whether the equation defines y as a function of $x$.

$y=x^2$

Determine whether the equation defines y as a function of $x$.

$y=x^3$

Determine whether the equation defines y as a function of $x$.

$y=\frac{1}{x}$

Determine whether the equation defines y as a function of $x$.

$y=\left|x\right|$

Determine whether the equation defines y as a function of $x$.

$y^2=4-x^2$

Determine whether the equation defines y as a function of $x$.

$y=\pm \sqrt{1-2x}$

Determine whether the equation defines y as a function of $x$.

$x=y^2$

Determine whether the equation defines y as a function of $x$.

$x+y^2=1$

Determine whether the equation defines y as a function of $x$.

$y=2x^2-3x+4$

Determine whether the equation defines y as a function of $x$.

$y=\frac{3x-1}{x+2}$

Determine whether the equation defines y as a function of $x$.

$2x^2+3y^2=1$

Determine whether the equation defines $y$ as a function of $x$.

$x^2-4y^2=1$

In Problems 39– 46, find the following for each function:

$\left(a\right) f\left(0\right) \left(b\right) f\left(1\right) \left(c\right) f(-1) \left(d\right) f(-x) \left(e\right) -f\left(x\right) \left(f\right) f(x+1) \left(g\right) f\left(x\right) \left(h\right) f(x+h)$

$f\left(x\right)=3x^2+2x-4$

In Problems 39– 46, find the following for each function:

$\left(a\right) f\left(0\right) \left(b\right) f\left(1\right) \left(c\right) f(-1) \left(d\right) f(-x) \left(e\right) -f\left(x\right) \left(f\right) f(x+1) \left(g\right) f\left(x\right) \left(h\right) f(x+h)$

$f\left(x\right)=-2x^2+x-1$

In Problems 39– 46, find the following for each function:

$\left(a\right) f\left(0\right) \left(b\right) f\left(1\right) \left(c\right) f(-1) \left(d\right) f(-x) \left(e\right) -f\left(x\right) \left(f\right) f(x+1) \left(g\right) f\left(x\right) \left(h\right) f(x+h)$

$f\left(x\right)=\frac{x}{x^2+1}$

In Problems 39– 46, find the following for each function:

$\left(a\right) f\left(0\right) \left(b\right) f\left(1\right) \left(c\right) f(-1) \left(d\right) f(-x) \left(e\right) -f\left(x\right) \left(f\right) f(x+1) \left(g\right) f\left(x\right) \left(h\right) f(x+h)$

$f\left(x\right)=\frac{x^2-1}{x+4}$

In Problems 39– 46, find the following for each function:

$\left(a\right) f\left(0\right) \left(b\right) f\left(1\right) \left(c\right) f(-1) \left(d\right) f(-x) \left(e\right) -f\left(x\right) \left(f\right) f(x+1) \left(g\right) f\left(x\right) \left(h\right) f(x+h)$

$f\left(x\right)=\left|x\right|+4$

In Problems 39– 46, find the following for each function:

$\left(a\right) f\left(0\right) \left(b\right) f\left(1\right) \left(c\right) f(-1) \left(d\right) f(-x) \left(e\right) -f\left(x\right) \left(f\right) f(x+1) \left(g\right) f\left(x\right) \left(h\right) f(x+h)$

$f\left(x\right)=\sqrt{x^2+x}$

In Problems 39– 46, find the following for each function:

$\left(a\right) f\left(0\right) \left(b\right) f\left(1\right) \left(c\right) f(-1) \left(d\right) f(-x) \left(e\right) -f\left(x\right) \left(f\right) f(x+1) \left(g\right) f\left(x\right) \left(h\right) f(x+h)$

$f\left(x\right)=\frac{2x+1}{3x-5}$'

In Problems 39– 46, find the following for each function:

$\left(a\right) f\left(0\right) \left(b\right) f\left(1\right) \left(c\right) f(-1) \left(d\right) f(-x) \left(e\right) -f\left(x\right) \left(f\right) f(x+1) \left(g\right) f\left(x\right) \left(h\right) f(x+h)$

$f\left(x\right)=1-\frac{1}{{(x+2)}^2}$

In Problems 47– 62, find the domain of each function.

$f\left(x\right)=-5x+4$