QuestionDetermine the range of the function $g(x) = x^{2}$ for the domain $-2 \leq x \leq 2$ .

Studdy Solution

STEP 1

Assumptions1. The function is $g(x)=x^{}$ . The domain of the function is $\{x \in R,- \leq x \leq\}$

STEP 2

The range of a function is the set of all possible output values (y-values), which are determined by the function's domain.For the given function $g(x)=x^{2}$ , we need to find the minimum and maximum values of $g(x)$ over the domain $-2 \leq x \leq2$ .

STEP 3

First, let's find the minimum value of $g(x)$ .
The function $g(x)=x^{2}$ is a parabola that opens upwards. This means that the minimum value of $g(x)$ occurs at the vertex of the parabola.

STEP 4

For the function $g(x)=x^{2}$ , the vertex is at $x=0$ . So, we substitute $x=0$ into the function to find the minimum value.
$g(0) = (0)^{2}$

STEP 5

Calculate the minimum value of $g(x)$ .
$g(0) = (0)^{2} =0$ So, the minimum value of $g(x)$ over the domain $-2 \leq x \leq2$ is0.

STEP 6

Now, let's find the maximum value of $g(x)$ .
Since the parabola opens upwards, the maximum value of $g(x)$ over the domain $-2 \leq x \leq2$ occurs at the endpoints of the domain, i.e., at $x=-2$ and $x=2$ .

STEP 7

Substitute $x=-2$ and $x=2$ into the function to find the maximum value.
$g(-2) = (-2)^{2}$ $g(2) = (2)^{2}$

STEP 8

Calculate the maximum value of $g(x)$ .
$g(-2) = (-2)^{2} =4$ $g(2) = (2)^{2} =4$ So, the maximum value of $g(x)$ over the domain $-2 \leq x \leq2$ is4.

STEP 9

Now that we have the minimum and maximum values of $g(x)$ over the domain $-2 \leq x \leq2$ , we can find the range of the function.
The range of the function is the set of all possible values of $g(x)$ , which is from the minimum value to the maximum value.
So, the range of the function $g(x)=x^{2}$ over the domain $-2 \leq x \leq2$ is $\{g(x) \in R, \leq g(x) \leq4\}$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

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

Parents Influencer program Contact Policy Terms