Domain and Range of Functions 2

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Question 1 of 4

1. Question
Find the domain and range of $y = - \frac{1}{x}$ $y=-1/x$
- 1. Domain: All real numbers $\neq 0$ $≠0$ ; Range: All real numbers $\neq 0$ $≠0$
- 2. Domain: All real numbers; Range: All real numbers
- 3. Domain: All positive numbers $\neq 0$ $≠0$ ; Range: All positive numbers $\neq 0$ $≠0$
- 4. Domain: All whole numbers $\neq 0$ $≠0$ ; Range: All whole numbers $\neq 0$ $≠0$
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The domain and range is the set of $x$ $x$ and $y$ $y$ values of a function

Notice that the curves extend infinitely on all sides but does not touch the $x$ $x$ and $y$ $y$ axis

This means that the curves will cover all real values of $x$ $x$ not equal to $0$ $0$ and all real values of $y$ $y$ not equal to $0$ $0$

$Domain: All real numbers \neq 0$ $\text(Domain: All real numbers)≠0$

$Range: All real numbers \neq 0$ $\text(Range: All real numbers)≠0$
Question 2 of 4

2. Question
Find the domain and range of $y = 4^{x}$ $y=4^x$
- 1. Domain: All real numbers; Range: $y > 0$ $y>0$
- 2. Domain: All real numbers; Range: All real numbers
- 3. Domain: All real numbers; Range: All real numbers $\neq 0$ $≠0$
- 4. Domain: $x > 0$ $x>0$ ; Range: All real numbers
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The domain and range is the set of $x$ $x$ and $y$ $y$ values of a function

Notice that the curve extends infinitely on both sides of the x-axis. It also extends upward infinitely, but does not touch the x-axis

This means that the curve will cover all real values of $x$ $x$ and all real values of $y$ $y$ greater than $0$ $0$

$Domain: All real numbers$ $\text(Domain: All real numbers)$

$Range : y > 0$ $\text(Range): y>0$
Question 3 of 4

3. Question
Find the domain and range

$f (x) = \frac{3}{x - 3}$ $f(x)=3/(x-3)$
- 1. Domain: $3$ $3$ ; Range: Factors of $3$ $3$
- 2. Domain: All real numbers except $3$ $3$ ; Range: All real numbers except $0$ $0$
- 3. Domain: $3$ $3$ ; Range: $0$ $0$
- 4. Domain: Factors of $3$ $3$ ; Range: All real numbers except $3$ $3$
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The domain and range is the set of $x$ $x$ and $y$ $y$ values of a function

Recall that fractions cannot have $0$ $0$ as the denominator or the value will be undefined

$f (x)$ $f(x)$ $=$ $=$ $\frac{3}{x - 3}$ $3/(x-3)$

$f (3)$ $f(3)$ $=$ $=$ $\frac{3}{3 - 3}$ $3/(3-3)$ Substitute $x = 3$ $x=3$

$=$ $=$ $\frac{3}{0}$ $3/0$

$=$ $=$ $undefined$ $\text(undefined)$

Since $x = 3$ $x=3$ makes the denominator equal to $0$ , $x≠3$

Therefore, the domain is all real numbers $≠3$

Next, recall that $f (x) = y$ $f(x)=y$ and $f (x)$ $f(x)$ cannot be equal to $0$ $0$ since the equation will have no solution.

Therefore, the range is all real numbers $≠0$

$\text(Domain: All real numbers)$ $≠3$

$\text(Range: All real numbers)$ $≠0$

Question 4 of 4

Graph and find the domain and range

f (x) = \frac{- 5}{x + 1}

$f(x)=(-5)/(x+1)$

1. Domain: All real numbers $< - 1$ $<-1$ ; Range: All real numbers $> 0$ $>0$
2. Domain: All real numbers; Range: All real numbers
3. Domain: All real numbers $\neq - 1$ $≠-1$ ; Range: All real numbers $\neq 0$ $≠0$
4. Domain: All real numbers $\neq - 1$ $≠-1$ ; Range: All real numbers $\neq 0$ $≠0$

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The domain and range is the set of $x$ $x$ and $y$ $y$ values of a function

First, use a table of values and test several values of

x

$x$ to get the value of

y

$y$

$x$ $x$	$- 5$ $-5$	$- 3$ $-3$	$- 1$ $-1$	$- \frac{1}{2}$ $-1/2$	$0$ $0$	$1$ $1$	$3$ $3$
$y$ $y$

Substitute the values of

x

$x$ to the function to get their

y

$y$ values

x = - 5

$x=-5$

$f (x)$ $f(x)$	$=$ $=$	$\frac{- 5}{x + 1}$ $(-5)/(x+1)$

$f (- 5)$ $f(-5)$	$=$ $=$	$\frac{- 5}{- 5 + 1}$ $(-5)/(-5+1)$	Substitute $x = - 5$ $x=-5$

	$=$ $=$	$- \frac{5}{- 4}$ $-5/-4$

	$=$ $=$	$1.3$ $1.3$	Rounded to one decimal place

x = - 3

$x=-3$

$f (x)$ $f(x)$	$=$ $=$	$\frac{- 5}{x + 1}$ $(-5)/(x+1)$

$f (- 3)$ $f(-3)$	$=$ $=$	$\frac{- 5}{- 3 + 1}$ $(-5)/(-3+1)$	Substitute $x = - 3$ $x=-3$

	$=$ $=$	$- \frac{5}{- 2}$ $-5/-2$

	$=$ $=$	$2.5$ $2.5$

x = - 1

$x=-1$

$f (x)$ $f(x)$	$=$ $=$	$\frac{- 5}{x + 1}$ $(-5)/(x+1)$

$f (- 1)$ $f(-1)$	$=$ $=$	$\frac{- 5}{- 1 + 1}$ $(-5)/(-1+1)$	Substitute $x = - 1$ $x=-1$

	$=$ $=$	$- \frac{5}{0}$ $-5/0$

	$=$ $=$	$undefined$ $\text(undefined)$

x = - \frac{1}{2}

$x=-1/2$

$f (x)$ $f(x)$	$=$ $=$	$\frac{- 5}{x + 1}$ $(-5)/(x+1)$

$f (- \frac{1}{2})$ $f(-1/2)$	$=$ $=$	$\frac{- 5}{- \frac{1}{2} + 1}$ $(-5)/(-1/2+1)$	Substitute $x = - \frac{1}{2}$ $x=-1/2$

	$=$ $=$	$- \frac{5}{\frac{1}{2}}$ $-5/(1/2)$

	$=$ $=$	$- 10$ $-10$

x = 0

$x=0$

$f (x)$ $f(x)$	$=$ $=$	$\frac{- 5}{x + 1}$ $(-5)/(x+1)$

$f (0)$ $f(0)$	$=$ $=$	$\frac{- 5}{0 + 1}$ $(-5)/(0+1)$	Substitute $x = 0$ $x=0$

	$=$ $=$	$- \frac{5}{1}$ $-5/1$

	$=$ $=$	$- 5$ $-5$

x = 1

$x=1$

$f (x)$ $f(x)$	$=$ $=$	$\frac{- 5}{x + 1}$ $(-5)/(x+1)$

$f (0)$ $f(0)$	$=$ $=$	$\frac{- 5}{1 + 1}$ $(-5)/(1+1)$	Substitute $x = 1$ $x=1$

	$=$ $=$	$- \frac{5}{2}$ $-5/2$

	$=$ $=$	$- 2.5$ $-2.5$

x = 3

$x=3$

$f (x)$ $f(x)$	$=$ $=$	$\frac{- 5}{x + 1}$ $(-5)/(x+1)$

$f (0)$ $f(0)$	$=$ $=$	$\frac{- 5}{3 + 1}$ $(-5)/(3+1)$	Substitute $x = 3$ $x=3$

	$=$ $=$	$- \frac{5}{4}$ $-5/4$

	$=$ $=$	$- 1.3$ $-1.3$	Rounded to one decimal place

$x$ $x$	$- 5$ $-5$	$- 3$ $-3$	$- 1$ $-1$	$- \frac{1}{2}$ $-1/2$	$0$ $0$	$1$ $1$	$3$
$y$	$1.3$	$2.5$	$\text(und.)$	$-10$	$-5$	$-2.5$	$-1.3$

Next, plot the points to the graph and connect them to form the curve.

Recall that undefined values are asymptotes

Notice that the curves extend infinitely on all sides but do not touch the asymptote at

$x=-1$ and the x-axis

This means that the curves will cover all real values of $x$ not equal to $-1$ and all real values of $y$ not equal to $0$

$\text(Domain: All real numbers)$

$≠-1$

$\text(Range: All real numbers)$

$≠0$

Domain and Range of Functions 2

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Quiz summary

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1. Question

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3. Question

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Quizzes

$f (x)$ $f(x)$	$=$ $=$	$\frac{3}{x - 3}$ $3/(x-3)$

$f (3)$ $f(3)$	$=$ $=$	$\frac{3}{3 - 3}$ $3/(3-3)$	Substitute $x = 3$ $x=3$

	$=$ $=$	$\frac{3}{0}$ $3/0$

	$=$ $=$	$undefined$ $\text(undefined)$