Math / Algebra

QuestionThe graph of an exponential function is shown in the figure below. The horizontal asymptote is shown as a dashed line. Find the range and the domain.
Write your answers as inequalities, using $x$ or $y$ as appropriate. Or, you may instead click on "Empty set" or "All reals" as the answer. (a) range: $\square$ $\square$ < $\square$ - $\square$ $\square$ $\leq \square$ (b) domain: $\square$

Studdy Solution

STEP 1

What is this asking? We need to find the set of all possible $y$ values (the range) and the set of all possible $x$ values (the domain) for this exponential function! Watch out! Don't mix up domain and range!
Remember, domain is about the $x$ values (left and right), and range is about the $y$ values (up and down).
Also, pay close attention to that horizontal asymptote!

STEP 2

1. Find the Domain
2. Find the Range

STEP 3

Imagine scanning the graph from left to right.
Does the graph keep going forever in both directions along the $x$ -axis?
Yes, it does!

STEP 4

This means the domain, the set of all possible $x$ values, includes every real number.

STEP 5

Now, imagine scanning the graph from bottom to top.
Does the graph keep going forever upwards?
No, it doesn't!
It seems to get really close to $y = 2$ , but it never actually touches it.

STEP 6

That dashed line at $y = 2$ is a horizontal asymptote.
The function gets closer and closer to $\mathbf{2}$ , but it never actually reaches it.
The graph does go upwards forever.

STEP 7

Since the graph is always above the asymptote, all the $y$ values are greater than $\mathbf{2}$ .
We can write this as $y > 2$ .

STEP 8

(a) range: $y > 2$ (b) domain: All reals

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Studdy Solution

STEP 1

STEP 2

1. Find the Domain
2. Find the Range

STEP 3

Imagine scanning the graph from left to right.
Does the graph keep going forever in both directions along the $x$ -axis?
Yes, it does!

STEP 4

This means the domain, the set of all possible $x$ values, includes every real number.

STEP 5

Now, imagine scanning the graph from bottom to top.
Does the graph keep going forever upwards?
No, it doesn't!
It seems to get really close to $y = 2$ , but it never actually touches it.

STEP 6

That dashed line at $y = 2$ is a horizontal asymptote.
The function gets closer and closer to $\mathbf{2}$ , but it never actually reaches it.
The graph does go upwards forever.

STEP 7

Since the graph is always above the asymptote, all the $y$ values are greater than $\mathbf{2}$ .
We can write this as $y > 2$ .

STEP 8

(a) range: $y > 2$ (b) domain: All reals

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Studdy Solution

STEP 1

STEP 2

1. Find the Domain2. Find the Range

STEP 3

Imagine scanning the graph from left to right.Does the graph keep going forever in both directions along the xxx-axis?Yes, it does!

STEP 4

This means the domain, the set of all possible xxx values, includes *every* real number.

STEP 5

Now, imagine scanning the graph from bottom to top.Does the graph keep going forever upwards?No, it doesn't!It seems to get really close to y=2y = 2y=2, but it never actually touches it.

STEP 6

That dashed line at y=2y = 2y=2 is a **horizontal asymptote**.The function gets closer and closer to 2\mathbf{2}2, but it never actually reaches it.The graph *does* go upwards forever.

STEP 7

Since the graph is always *above* the asymptote, all the yyy values are *greater than* 2\mathbf{2}2.We can write this as y>2y > 2y>2.

STEP 8

(a) range: y>2y > 2y>2 (b) domain: All reals

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Studdy solves anything!

Start learning now

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

1. Find the Domain
2. Find the Range

Imagine scanning the graph from left to right.
Does the graph keep going forever in both directions along the $x$ -axis?
Yes, it does!

This means the domain, the set of all possible $x$ values, includes every real number.

Now, imagine scanning the graph from bottom to top.
Does the graph keep going forever upwards?
No, it doesn't!
It seems to get really close to $y = 2$ , but it never actually touches it.

That dashed line at $y = 2$ is a horizontal asymptote.
The function gets closer and closer to $\mathbf{2}$ , but it never actually reaches it.
The graph does go upwards forever.

Since the graph is always above the asymptote, all the $y$ values are greater than $\mathbf{2}$ .
We can write this as $y > 2$ .

(a) range: $y > 2$ (b) domain: All reals