Solved on Nov 07, 2023

Graph the function $h(x) = -\frac{1}{x^2}$ . Find its domain, $x$ -intercepts, $y$ -intercept, and vertical asymptotes.
The domain of the function is $\mathbb{R} \setminus \{0\}$ . The $x$ -intercept(s) is/are $(0,0)$ . The $y$ -intercept is $(0,-1)$ . The function has one vertical asymptote at $x = 0$ .

STEP 1

Assumptions1. The function we are working with is $h(x)=-\frac{1}{x^{}}$ . . We are asked to graph the function, determine the asymptotes, list the domain, and find the $x$ -intercepts and $y$ -intercepts.
3. We will use a graphing calculator to check our work.

STEP 2

First, let's find the domain of the function. The domain of a function is the set of all possible $x$ -values. For this function, $x$ can be any real number except for $0$ because division by $0$ is undefined.
$Domain = (-\infty,0) \cup (0, \infty)$

STEP 3

Next, let's find the $x$ -intercepts of the function. The $x$ -intercepts are the values of $x$ where $h(x) =0$ . To find these, we set $h(x)$ equal to $0$ and solve for $x$ .
$0 = -\frac{1}{x^{2}}$

STEP 4

We can see that there are no real solutions for $x$ in the equation from3, so the function has no $x$ -intercepts.

STEP 5

Now, let's find the $y$ -intercept of the function. The $y$ -intercept is the value of $h(x)$ when $x =0$ . However, as we noted when finding the domain, $x$ cannot be $0$ for this function. Therefore, the function has no $y$ -intercept.

STEP 6

Next, let's find the asymptotes of the function. An asymptote is a line that the graph of the function approaches but never touches. For this function, there is a vertical asymptote at $x =0$ because the function is undefined at $x =0$ .
$x =0$

STEP 7

Now, let's graph the function using a graphing calculator. We should see a graph that is a downward opening hyperbola with a vertical asymptote at $x =0$ .

STEP 8

Finally, let's check our work using a graphing calculator. The graph should match our description from7, and the calculator should confirm that the domain is $(-\infty,0) \cup (0, \infty)$ , there are no $x$ -intercepts or $y$ -intercepts, and the vertical asymptote is $x =0$ .

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STEP 1

Assumptions1. The function we are working with is $h(x)=-\frac{1}{x^{}}$ . . We are asked to graph the function, determine the asymptotes, list the domain, and find the $x$ -intercepts and $y$ -intercepts.
3. We will use a graphing calculator to check our work.

STEP 2

First, let's find the domain of the function. The domain of a function is the set of all possible $x$ -values. For this function, $x$ can be any real number except for $0$ because division by $0$ is undefined.
$Domain = (-\infty,0) \cup (0, \infty)$

STEP 3

Next, let's find the $x$ -intercepts of the function. The $x$ -intercepts are the values of $x$ where $h(x) =0$ . To find these, we set $h(x)$ equal to $0$ and solve for $x$ .
$0 = -\frac{1}{x^{2}}$

STEP 4

We can see that there are no real solutions for $x$ in the equation from3, so the function has no $x$ -intercepts.

STEP 5

Now, let's find the $y$ -intercept of the function. The $y$ -intercept is the value of $h(x)$ when $x =0$ . However, as we noted when finding the domain, $x$ cannot be $0$ for this function. Therefore, the function has no $y$ -intercept.

STEP 6

Next, let's find the asymptotes of the function. An asymptote is a line that the graph of the function approaches but never touches. For this function, there is a vertical asymptote at $x =0$ because the function is undefined at $x =0$ .
$x =0$

STEP 7

Now, let's graph the function using a graphing calculator. We should see a graph that is a downward opening hyperbola with a vertical asymptote at $x =0$ .

STEP 8

Finally, let's check our work using a graphing calculator. The graph should match our description from7, and the calculator should confirm that the domain is $(-\infty,0) \cup (0, \infty)$ , there are no $x$ -intercepts or $y$ -intercepts, and the vertical asymptote is $x =0$ .

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STEP 1

Assumptions1. The function we are working with is h(x)=−1xh(x)=-\frac{1}{x^{}}h(x)=−x1​. . We are asked to graph the function, determine the asymptotes, list the domain, and find the xxx-intercepts and yyy-intercepts.3. We will use a graphing calculator to check our work.

STEP 2

First, let's find the domain of the function. The domain of a function is the set of all possible xxx-values. For this function, xxx can be any real number except for 000 because division by 000 is undefined.Domain=(−∞,0)∪(0,∞)Domain = (-\infty,0) \cup (0, \infty)Domain=(−∞,0)∪(0,∞)

STEP 3

Next, let's find the xxx-intercepts of the function. The xxx-intercepts are the values of xxx where h(x)=0h(x) =0h(x)=0. To find these, we set h(x)h(x)h(x) equal to 000 and solve for xxx.0=−1x20 = -\frac{1}{x^{2}}0=−x21​

STEP 4

We can see that there are no real solutions for xxx in the equation from3, so the function has no xxx-intercepts.

STEP 5

Now, let's find the yyy-intercept of the function. The yyy-intercept is the value of h(x)h(x)h(x) when x=0x =0x=0. However, as we noted when finding the domain, xxx cannot be 000 for this function. Therefore, the function has no yyy-intercept.

STEP 6

Next, let's find the asymptotes of the function. An asymptote is a line that the graph of the function approaches but never touches. For this function, there is a vertical asymptote at x=0x =0x=0 because the function is undefined at x=0x =0x=0.x=0x =0x=0

STEP 7

Now, let's graph the function using a graphing calculator. We should see a graph that is a downward opening hyperbola with a vertical asymptote at x=0x =0x=0.

STEP 8

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Assumptions1. The function we are working with is $h(x)=-\frac{1}{x^{}}$ . . We are asked to graph the function, determine the asymptotes, list the domain, and find the $x$ -intercepts and $y$ -intercepts.
3. We will use a graphing calculator to check our work.

First, let's find the domain of the function. The domain of a function is the set of all possible $x$ -values. For this function, $x$ can be any real number except for $0$ because division by $0$ is undefined.
$Domain = (-\infty,0) \cup (0, \infty)$

Next, let's find the $x$ -intercepts of the function. The $x$ -intercepts are the values of $x$ where $h(x) =0$ . To find these, we set $h(x)$ equal to $0$ and solve for $x$ .
$0 = -\frac{1}{x^{2}}$

We can see that there are no real solutions for $x$ in the equation from3, so the function has no $x$ -intercepts.

Now, let's find the $y$ -intercept of the function. The $y$ -intercept is the value of $h(x)$ when $x =0$ . However, as we noted when finding the domain, $x$ cannot be $0$ for this function. Therefore, the function has no $y$ -intercept.

Next, let's find the asymptotes of the function. An asymptote is a line that the graph of the function approaches but never touches. For this function, there is a vertical asymptote at $x =0$ because the function is undefined at $x =0$ .
$x =0$

Now, let's graph the function using a graphing calculator. We should see a graph that is a downward opening hyperbola with a vertical asymptote at $x =0$ .