Section 1
Functions and Function Notation
What is the difference between a relation and a function?
Determine whether the relation represents $y$ as a function of $x$. $y=\frac{1}{x}$
What is the difference between the input and the output of a function?
Given the function $g(x)=x^2+2 x$, evaluate $\frac{g(x)-g(a)}{x-a}, x \neq a$.
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 oneto-one?
Determine whether the relation represents a function. $\{(a, b),(c, d),(a, c)\}$
Determine whether the relation represents a function. $\{(a, b),(b, c),(c, c)\}$
Determine whether the relation represents $y$ as a function of $x$. $5 x+2 y=10$
Determine whether the relation represents $y$ as a function of $x$. $y=x^2$
Determine whether the relation represents $y$ as a function of $x$. $x=y^2$
Determine whether the relation represents $y$ as a function of $x$. $3 x^2+y=14$
Determine whether the relation represents $y$ as a function of $x$. $2 x+y^2=6$
Determine whether the relation represents $y$ as a function of $x$. $y=-2 x^2+40 x$
Determine whether the relation represents $y$ as a function of $x$. $x=\frac{3 y+5}{7 y-1}$
Determine whether the relation represents $y$ as a function of $x$. $x=\sqrt{1-y^2}$
Determine whether the relation represents $y$ as a function of $x$. $y=\frac{3 x+5}{7 x-1}$
Determine whether the relation represents $y$ as a function of $x$. $x^2+y^2=9$
Determine whether the relation represents $y$ as a function of $x$. $2 x y=1$
Determine whether the relation represents $y$ as a function of $x$. $x=y^3$
Determine whether the relation represents $y$ as a function of $x$. $y=x^3$
Determine whether the relation represents $y$ as a function of $x$. $y=\sqrt{1-x^2}$
Determine whether the relation represents $y$ as a function of $x$. $x= \pm \sqrt{1-y}$
Determine whether the relation represents $y$ as a function of $x$. $y= \pm \sqrt{1-x}$
Determine whether the relation represents $y$ as a function of $x$. $y^2=x^2$
Determine whether the relation represents $y$ as a function of $x$. $y^3=x^2$
Evaluate $f(-3), f(2), f(-a),-f(a), f(a+h)$. $f(x)=2 x-5$
Evaluate $f(-3), f(2), f(-a),-f(a), f(a+h)$. $f(x)=-5 x^2+2 x-1$
Evaluate $f(-3), f(2), f(-a),-f(a), f(a+h)$. $f(x)=\sqrt{2-x}+5$
.Evaluate $f(-3), f(2), f(-a),-f(a), f(a+h)$. $f(x)=\frac{6 x-1}{5 x+2}$
Evaluate $f(-3), f(2), f(-a),-f(a), f(a+h)$. $f(x)=|x-1|-|x+1|$
Given the function $g(x)=5-x^2$, evaluate $\frac{g(x+h)-g(x)}{h}, h \neq 0$.
Given the function $p(c)=c^2+c:$ (a) Evaluate $p(-3)$. (b) Solve $p(c)=2$.
. Given the function $k(t)=2 t-1:$ (a) Evaluate $k(2)$. (b) Solve $k(t)=7$.
Given the function $f(x)=x^2-3 x:$ (a) Evaluate $f(5)$. (b) Solve $f(x)=4$.
Given the function $f(x)=8-3 x:$ (a) Evaluate $f(-2)$. (b) Solve $f(x)=-1$.
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 $f(t)=2$.
Use the vertical line test to determine which graphs show relations that are functions.
GRAPH CAN'T COPY
Given the following graph,(a) Evaluate $f(-1)$.(b) Solve for $f(x)=3$.
Given the following graph,(a) Evaluate $f(0)$.(b) Solve for $f(x)=-3$.
Given the following graph,(a) Evaluate $f(4)$.(b) Solve for $f(x)=1$.
Determine if the given graph is a one-to-one function.
determine whether the relation represents a function. $\{(-1,-1),(-2,-2),(-3,-3)\}$
determine whether the relation represents a function. $\{(3,4),(4,5),(5,6)\}$
determine whether the relation represents a function. $\{(2,5),(7,11),(15,8),(7,9)\}$
Determine if the relation represented in table form represents $y$ as a function of $x$.
$$\begin{array}{|c|c|c|c|}\hline x & 5 & 10 & 15 \\\hline y & 3 & 8 & 14 \\\hline\end{array}$$
$$\begin{array}{|c|c|c|c|}\hline x & 5 & 10 & 15 \\\hline y & 3 & 8 & 8 \\\hline\end{array}$$
Determine if the relation represented in table form represents as a function of X$$\begin{array}{|c|c|c|c|}\hline x & 5 & 10 & 10 \\\hline y & 3 & 8 & 14 \\\hline\end{array}$$
$$\text { Use the function } f \text { represented in the table below. }$$$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}\hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\\hline f(x) & 74 & 28 & 1 & 53 & 56 & 3 & 36 & 45 & 14 & 47 \\\hline\end{array}$$
Evaluate $f(3)$.
Solve $f(x)=1$.
Evaluate the function $f$ at the values $f(-2), f(-1), f(0), f(1)$, and $f(2)$. $f(x)=4-2 x$
Evaluate the function $f$ at the values $f(-2), f(-1), f(0), f(1)$, and $f(2)$. $f(x)=8-3 x$
Evaluate the function $f$ at the values $f(-2), f(-1), f(0), f(1)$, and $f(2)$. $f(x)=8 x^2-7 x+3$
Evaluate the function $f$ at the values $f(-2), f(-1), f(0), f(1)$, and $f(2)$. $f(x)=3+\sqrt{x+3}$
Evaluate the function $f$ at the values $f(-2), f(-1), f(0), f(1)$, and $f(2)$. $f(x)=\frac{x-2}{x+3}$
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 $h$ :
$$\begin{aligned}& f(x)=3 x-2 \\& g(x)=5-x^2 \\& h(x)=-2 x^2+3 x-1\end{aligned}$$
$3 f(1)-4 g(-2)$
$f\left(\frac{7}{3}\right)-h(-2)$
Determine the corresponding range. Show each graph. $[-0.1,0.1]$
Determine the corresponding range. Show each graph. $[-10,10]$
Determine the corresponding range. Show each graph. $[-100,100]$
Determine the corresponding range. Show each graph. $[0,0.01]$
Determine the corresponding range. Show each graph. $[0,100]$
Determine the corresponding range. Show each graph. $[0,10,000]$
Determine the corresponding range. Show each graph. $[-0.001,0.001]$
Determine the corresponding range. Show each graph. $[-1000,1000]$
Determine the corresponding range. Show each graph. $[-1,000,000,1,000,000]$
The amount of garbage, $G$, produced by a city with population $p$ is given by $G=f(p) \cdot G$ is measured in tons per week, and $p$ is measured in thousands of people.(a) The town of Tola has apopulation 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 $5000 \mathrm{ft}^2$ 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 $h(t)$ 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.
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) $f(10)=40$