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Section 1
Functions and Function Notation
What is the difference between a relation and a function?
What is the difference between the input and the output of a function?
Why does the vertical line test tell us whether the graph of a relation represents a function?
How can you determine if a relation is a one-to-one function?
Why does the horizontal line test tell us whether the graph of a function is one-to-one?
For the following exercises, determine whether the relation represents a function.$$\{(a, b),(c, d),(a, c)\}$$
For the following exercises, determine whether the relation represents a function.$$\{(a, b),(b, c),(c, c)\}$$
For the following exercises, determine whether the relation represents $y$ as a function of $x .$$$5 x+2 y=10$$
For the following exercises, determine whether the relation represents $y$ as a function of $x .$$$y=x^{2}$$
For the following exercises, determine whether the relation represents $y$ as a function of $x .$$$x=y^{2}$$
For the following exercises, determine whether the relation represents $y$ as a function of $x .$$$3 x^{2}+y=14$$
For the following exercises, determine whether the relation represents $y$ as a function of $x .$$$2 x+y^{2}=6$$
For the following exercises, determine whether the relation represents $y$ as a function of $x .$$$y=-2 x^{2}+40 x$$
For the following exercises, determine whether the relation represents $y$ as a function of $x .$$$y=\frac{1}{x}$$
For the following exercises, determine whether the relation represents $y$ as a function of $x .$$$x=\frac{3 y+5}{7 y-1}$$
For the following exercises, determine whether the relation represents $y$ as a function of $x .$$$x=\sqrt{1-y^{2}}$$
For the following exercises, determine whether the relation represents $y$ as a function of $x .$$$y=\frac{3 x+5}{7 x-1}$$
For the following exercises, determine whether the relation represents $y$ as a function of $x .$$$x^{2}+y^{2}=9$$
For the following exercises, determine whether the relation represents $y$ as a function of $x .$$$2 x y=1$$
For the following exercises, determine whether the relation represents $y$ as a function of $x .$$$x=y^{3}$$
For the following exercises, determine whether the relation represents $y$ as a function of $x .$$$y=x^{3}$$
For the following exercises, determine whether the relation represents $y$ as a function of $x .$$$y=\sqrt{1-x^{2}}$$
For the following exercises, determine whether the relation represents $y$ as a function of $x .$$$x=\pm \sqrt{1-y}$$
For the following exercises, determine whether the relation represents $y$ as a function of $x .$$$y=\pm \sqrt{1-x}$$
For the following exercises, determine whether the relation represents $y$ as a function of $x .$$$y^{2}=x^{2}$$
For the following exercises, determine whether the relation represents $y$ as a function of $x .$$$y^{3}=x^{2}$$
For the following exercises, evaluate the function $f$ at the indicated values $f(-3), f(2), f(-a),-f(a), f(a+h)$.$$f(x)=2 x-5$$
For the following exercises, evaluate the function $f$ at the indicated values $f(-3), f(2), f(-a),-f(a), f(a+h)$.$$f(x)=-5 x^{2}+2 x-1$$
For the following exercises, evaluate the function $f$ at the indicated values $f(-3), f(2), f(-a),-f(a), f(a+h)$.$$f(x)=\sqrt{2-x}+5$$
For the following exercises, evaluate the function $f$ at the indicated values $f(-3), f(2), f(-a),-f(a), f(a+h)$.$$f(x)=\frac{6 x-1}{5 x+2}$$
For the following exercises, evaluate the function $f$ at the indicated values $f(-3), f(2), f(-a),-f(a), f(a+h)$.$$f(x)=|x-1|-|x+1|$$
Given the function $g(x)=5-x^{2},$ simplify $\frac{g(x+h)-g(x)}{h}, h \neq 0$
Given the function $g(x)=x^{2}+2 x,$ simplify $\frac{g(x)-g(a)}{x-a}, x \neq a$
Given the function $k(t)=2 t-1$a. Evaluate $k(2)$b. Solve $k(t)=7$
Given the function $f(x)=8-3 x$a. Evaluate $f(-2)$b. Solve $f(x)=-1$
Given the function $p(c)=c^{2}+c$a. Evaluate $p(-3)$b. Solve $p(c)=2$
Given the function $f(x)=x^{2}-3 x$a. Evaluate $f(5)$b. Solve $f(x)=4$
Given the function $f(x)=\sqrt{x+2}$a. Evaluate $f(7)$b. Solve $f(x)=4$
Consider the relationship $3 r+2 t=18$a. Write the relationship as a function $r=f(t)$b. Evaluate $f(-3)$c. Solve $f(t)=2$
For the following exercises, use the vertical line test to determine which graphs show relations that are functions.
Given the following grapha. Evaluate $f(-1)$b. Solve for $f(x)=3$
Given the following grapha. Evaluate $f(0)$b. Solve for $f(x)=-3$
Given the following grapha. Evaluate $f(4)$b. Solve for $f(x)=1$
For the following exercises, determine if the given graph is a one-to-one function.
For the following exercises, determine whether the relation represents a function.$$\{(-1,-1),(-2,-2),(-3,-3)\}$$
For the following exercises, determine whether the relation represents a function.$$\{(3,4),(4,5),(5,6)\}$$
For the following exercises, determine whether the relation represents a function.$$\{(2,5),(7,11),(15,8),(7,9)\}$$
For the following exercises, determine if the relation represented in table form represents $y$ as a function of $x .$$$\begin{array}{|c|c|c|c|}\hline \boldsymbol{x} & 5 & 10 & 15 \\\hline \boldsymbol{y} & 3 & 8 & 14 \\\hline\end{array}$$
For the following exercises, determine if the relation represented in table form represents $y$ as a function of $x .$$$\begin{array}{|c|c|c|c|}\hline x & 5 & 10 & 15 \\\hline y & 3 & 8 & 8 \\\hline\end{array}$$
For the following exercises, determine if the relation represented in table form represents $y$ as a function of $x .$$$\begin{array}{|c|c|c|c|}\hline x & 5 & 10 & 10 \\\hline y & 3 & 8 & 14 \\\hline\end{array}$$
For the following exercises, use the function $f$ represented in Table 14 below.$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}\hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\\hline f(x) & 74 & 28 & 1 & 53 & 56 & 3 & 36 & 45 & 14 & 47 \\\hline\end{array}$$Evaluate $f(3)$
For the following exercises, use the function $f$ represented in Table 14 below.$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}\hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\\hline f(x) & 74 & 28 & 1 & 53 & 56 & 3 & 36 & 45 & 14 & 47 \\\hline\end{array}$$Solve $f(x)=1$
For the following exercises, evaluate the function $f$ at the values $f(-2), f(-1), f(0), f(1),$ and $f(2)$.$$f(x)=4-2 x$$
For the following exercises, evaluate the function $f$ at the values $f(-2), f(-1), f(0), f(1),$ and $f(2)$.$$f(x)=8-3 x$$
For the following exercises, evaluate the function $f$ at the values $f(-2), f(-1), f(0), f(1),$ and $f(2)$.$$f(x)=8 x^{2}-7 x+3$$
For the following exercises, evaluate the function $f$ at the values $f(-2), f(-1), f(0), f(1),$ and $f(2)$.$$f(x)=3+\sqrt{x+3}$$
For the following exercises, evaluate the function $f$ at the values $f(-2), f(-1), f(0), f(1),$ and $f(2)$.$$f(x)=\frac{x-2}{x+3}$$
For the following exercises, evaluate the function $f$ at the values $f(-2), f(-1), f(0), f(1),$ and $f(2)$.$$f(x)=3^{x}$$
For the following exercises, evaluate the expressions, given functions $f, g,$ and $h:$$$f(x)=3 x-2 \quad g(x)=5-x^{2} \quad h(x)=-2 x^{2}+3 x-1$$$$3 f(1)-4 g(-2)$$
For the following exercises, evaluate the expressions, given functions $f, g,$ and $h:$$$f(x)=3 x-2 \quad g(x)=5-x^{2} \quad h(x)=-2 x^{2}+3 x-1$$$$f\left(\frac{7}{3}\right)-h(-2)$$
For the following exercises, graph $y=x^{2}$ on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.$$[-0.1,0.1]$$
For the following exercises, graph $y=x^{2}$ on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.$$[-10,10]$$
For the following exercises, graph $y=x^{2}$ on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.$$[-100,100]$$
For the following exercises, graph $y=x^{3}$ on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.$$[-0.1,0.1]$$
For the following exercises, graph $y=x^{3}$ on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.$$[-10,10]$$
For the following exercises, graph $y=x^{3}$ on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.$$[-100,100]$$
For the following exercises, graph $y=\sqrt{x}$ on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.$$[0,0.01]$$
For the following exercises, graph $y=\sqrt{x}$ on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.$$[0,100]$$
For the following exercises, graph $y=\sqrt{x}$ on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.$$[0,10,000]$$
For the following exercises, graph $y=\sqrt[3]{x}$ on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.$$[-0.001,0.001]$$
For the following exercises, graph $y=\sqrt[3]{x}$ on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.$$[-1,000,1,000]$$
For the following exercises, graph $y=\sqrt[3]{x}$ on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.$$[-1,000,000,1,000,000]$$
The amount of garbage, $G$, produced by a city with population $p$ is given by $G=f(p) . G$ is measured in tons per week, and $p$ is measured in thousands of people.a. The town of Tola has a population of 40,000 and produces 13 tons of garbage each week. Express this information in terms of the function $f$b. Explain the meaning of the statement $f(5)=2$
The number of cubic yards of dirt, $D$, needed to cover a garden with area $a$ square feet is given by $D=g(a)$a. A garden with area $5,000 \mathrm{ft}^{2}$ requires $50 \mathrm{yd}^{3}$ of dirt. Express this information in terms of the function $g$.b. Explain the meaning of the statement $g(100)=1$
Let $f(t)$ be the number of ducks in a lake $t$ years after 1990. Explain the meaning of each statement:a. $f(5)=30$b. $f(10)=40$
Let $h(t)$ be the height above ground, in feet, of a rocket $t$ seconds after launching. Explain the meaning of each statement:a. $h(1)=200$b. $h(2)=350$
Show that the function $f(x)=3(x-5)^{2}+7$ is not one-to-one.