QuestionFind the range of the function $f(x) = \begin{cases} 0 & \text{if } x \leq -4 \\ -x & \text{if } -4 < x < 0 \\ x^2 & \text{if } x \geq 0 \end{cases}$ .

Studdy Solution

STEP 1

Assumptions1. The function is piece-wise defined with three parts $0$ for $x \leq -4$ , $-x$ for $-4 \leq x <0$ , and $x^{}$ for $x \geq0$ .

STEP 2

First, we need to determine the range of each piece of the function separately.For $x \leq -4$ , the function is constant at $0$ . So, the range for this part is just $0$ .

STEP 3

For $- \leq x <0$ , the function is $-x$ . As $x$ increases from $-$ to $0$ , $-x$ decreases from $$ to $0$. So, the range for this part is $[0,)$.

STEP 4

For $x \geq0$ , the function is $x^{2}$ . As $x$ increases from $0$ to $\infty$ , $x^{2}$ also increases from $0$ to $\infty$ . So, the range for this part is $[0, \infty)$ .

STEP 5

Now, we combine the ranges of all the pieces to get the range of the entire function.
The range of the function $f(x)$ is the set of all possible values of $f(x)$ , which is the union of the ranges of all the pieces.
So, the range of $f(x)$ is $[0, \infty)$ .

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QuestionFind the range of the function $f(x) = \begin{cases} 0 & \text{if } x \leq -4 \\ -x & \text{if } -4 < x < 0 \\ x^2 & \text{if } x \geq 0 \end{cases}$ .

Studdy Solution

STEP 1

Assumptions1. The function is piece-wise defined with three parts $0$ for $x \leq -4$ , $-x$ for $-4 \leq x <0$ , and $x^{}$ for $x \geq0$ .

STEP 2

First, we need to determine the range of each piece of the function separately.For $x \leq -4$ , the function is constant at $0$ . So, the range for this part is just $0$ .

STEP 3

For $- \leq x <0$ , the function is $-x$ . As $x$ increases from $-$ to $0$ , $-x$ decreases from $$ to $0$. So, the range for this part is $[0,)$.

STEP 4

For $x \geq0$ , the function is $x^{2}$ . As $x$ increases from $0$ to $\infty$ , $x^{2}$ also increases from $0$ to $\infty$ . So, the range for this part is $[0, \infty)$ .

STEP 5

Now, we combine the ranges of all the pieces to get the range of the entire function.
The range of the function $f(x)$ is the set of all possible values of $f(x)$ , which is the union of the ranges of all the pieces.
So, the range of $f(x)$ is $[0, \infty)$ .

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QuestionFind the range of the function f(x)={0if x≤−4−xif −4<x<0x2if x≥0f(x) = \begin{cases} 0 & \text{if } x \leq -4 \\ -x & \text{if } -4 < x < 0 \\ x^2 & \text{if } x \geq 0 \end{cases}f(x)=⎩⎨⎧​0−xx2​if x≤−4if −4<x<0if x≥0​.

Studdy Solution

STEP 1

Assumptions1. The function is piece-wise defined with three parts 000 for x≤−4x \leq -4x≤−4, −x-x−x for −4≤x<0-4 \leq x <0−4≤x<0, and xx^{}x for x≥0x \geq0x≥0.

STEP 2

First, we need to determine the range of each piece of the function separately.For x≤−4x \leq -4x≤−4, the function is constant at 000. So, the range for this part is just 000.

STEP 3

For −≤x<0- \leq x <0−≤x<0, the function is −x-x−x. As xxx increases from −-− to 000, −x-x−x decreases from $$ to $0$. So, the range for this part is $[0,)$.

STEP 4

For x≥0x \geq0x≥0, the function is x2x^{2}x2. As xxx increases from 000 to ∞\infty∞, x2x^{2}x2 also increases from 000 to ∞\infty∞. So, the range for this part is [0,∞)[0, \infty)[0,∞).

STEP 5

Now, we combine the ranges of all the pieces to get the range of the entire function.The range of the function f(x)f(x)f(x) is the set of all possible values of f(x)f(x)f(x), which is the union of the ranges of all the pieces.So, the range of f(x)f(x)f(x) is [0,∞)[0, \infty)[0,∞).

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QuestionFind the range of the function $f(x) = \begin{cases} 0 & \text{if } x \leq -4 \\ -x & \text{if } -4 < x < 0 \\ x^2 & \text{if } x \geq 0 \end{cases}$ .

Assumptions1. The function is piece-wise defined with three parts $0$ for $x \leq -4$ , $-x$ for $-4 \leq x <0$ , and $x^{}$ for $x \geq0$ .

First, we need to determine the range of each piece of the function separately.For $x \leq -4$ , the function is constant at $0$ . So, the range for this part is just $0$ .

For $- \leq x <0$ , the function is $-x$ . As $x$ increases from $-$ to $0$ , $-x$ decreases from $$ to $0$. So, the range for this part is $[0,)$.

For $x \geq0$ , the function is $x^{2}$ . As $x$ increases from $0$ to $\infty$ , $x^{2}$ also increases from $0$ to $\infty$ . So, the range for this part is $[0, \infty)$ .

Now, we combine the ranges of all the pieces to get the range of the entire function.
The range of the function $f(x)$ is the set of all possible values of $f(x)$ , which is the union of the ranges of all the pieces.
So, the range of $f(x)$ is $[0, \infty)$ .