QuestionFind the domain of $f(x)=\left\{\begin{array}{ll}5 x & \text { if }-1 \leq x<3 \\ x+4 & \text { if } 3 \leq x \leq 4\end{array}\right.$ , evaluate $f(-2), f(0), f(5)$ , graph $f$ , and check continuity.

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is defined as $f(x)=\left\{\begin{array}{ll}5 x & \text { if }-1 \leq x<3 \\ x+4 & \text { if }3 \leq x \leq4\end{array}\right.$ . We need to determine the domain of $f$ , evaluate $f(-), f(0)$ , and $f(5)$ , graph $f$ , and determine if $f$ is continuous on its domain.

STEP 2

To determine the domain of $f$ , we need to consider the values of $x$ for which $f(x)$ is defined. From the definition of $f(x)$ , we can see that $f(x)$ is defined for $-1 \leq x <$ and $\leq x \leq4$ .So the domain of $f$ is $-1 \leq x \leq4$ .

STEP 3

Now, let's evaluate $f(-2), f(0)$ , and $f(5)$ .For $f(-2)$ , we see that $-2$ is not in the domain of $f$ (since $-1 \leq x \leq$ ), so $f(-2)$ is undefined.

STEP 4

For $f(0)$ , we see that $0$ is in the first part of the domain of $f$ (since $-1 \leq x <3$ ). So, we use the first part of the function definition to evaluate $f(0)$ .
$f(0) = \times0 =0$

STEP 5

For $f(5)$ , we see that $5$ is not in the domain of $f$ (since $-1 \leq x \leq4$ ), so $f(5)$ is undefined.

STEP 6

To graph $f$ , we plot the function for the two parts of the domain separately. For $-1 \leq x <3$ , we plot the line $y =5x$ . For $3 \leq x \leq4$ , we plot the line $y = x +4$ .

STEP 7

To determine if $f$ is continuous on its domain, we need to check if the function is continuous at the point where the two parts of the function definition meet, which is $x =3$ .
At $x =3$ , from the first part of the function definition, $f(3) =5 \times3 =15$ . And from the second part of the function definition, $f(3) =3 +4 =7$ . Since $15 \neq7$ , $f$ is not continuous at $x =3$ .
So, $f$ is not continuous on its domain.

STEP 8

Summarizing the results(a) The domain of $f$ is $-1 \leq x \leq4$ . (b) $f(-2)$ is undefined, $f(0) =0$ , and $f(5)$ is undefined. (c) The graph of $f$ consists of two lines $y =5x$ for $-1 \leq x <3$ and $y = x +4$ for $3 \leq x \leq4$ . (d) $f$ is not continuous on its domain.

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QuestionFind the domain of $f(x)=\left\{\begin{array}{ll}5 x & \text { if }-1 \leq x<3 \\ x+4 & \text { if } 3 \leq x \leq 4\end{array}\right.$ , evaluate $f(-2), f(0), f(5)$ , graph $f$ , and check continuity.

Studdy Solution

STEP 1

STEP 2

To determine the domain of $f$ , we need to consider the values of $x$ for which $f(x)$ is defined. From the definition of $f(x)$ , we can see that $f(x)$ is defined for $-1 \leq x <$ and $\leq x \leq4$ .So the domain of $f$ is $-1 \leq x \leq4$ .

STEP 3

Now, let's evaluate $f(-2), f(0)$ , and $f(5)$ .For $f(-2)$ , we see that $-2$ is not in the domain of $f$ (since $-1 \leq x \leq$ ), so $f(-2)$ is undefined.

STEP 4

For $f(0)$ , we see that $0$ is in the first part of the domain of $f$ (since $-1 \leq x <3$ ). So, we use the first part of the function definition to evaluate $f(0)$ .
$f(0) = \times0 =0$

STEP 5

For $f(5)$ , we see that $5$ is not in the domain of $f$ (since $-1 \leq x \leq4$ ), so $f(5)$ is undefined.

STEP 6

To graph $f$ , we plot the function for the two parts of the domain separately. For $-1 \leq x <3$ , we plot the line $y =5x$ . For $3 \leq x \leq4$ , we plot the line $y = x +4$ .

STEP 7

STEP 8

Summarizing the results(a) The domain of $f$ is $-1 \leq x \leq4$ . (b) $f(-2)$ is undefined, $f(0) =0$ , and $f(5)$ is undefined. (c) The graph of $f$ consists of two lines $y =5x$ for $-1 \leq x <3$ and $y = x +4$ for $3 \leq x \leq4$ . (d) $f$ is not continuous on its domain.

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Studdy Solution

STEP 1

STEP 2

STEP 3

Now, let's evaluate f(−2),f(0)f(-2), f(0)f(−2),f(0), and f(5)f(5)f(5).For f(−2)f(-2)f(−2), we see that −2-2−2 is not in the domain of fff (since −1≤x≤-1 \leq x \leq−1≤x≤), so f(−2)f(-2)f(−2) is undefined.

STEP 4

For f(0)f(0)f(0), we see that 000 is in the first part of the domain of fff (since −1≤x<3-1 \leq x <3−1≤x<3). So, we use the first part of the function definition to evaluate f(0)f(0)f(0).f(0)=×0=0f(0) = \times0 =0f(0)=×0=0

STEP 5

For f(5)f(5)f(5), we see that 555 is not in the domain of fff (since −1≤x≤4-1 \leq x \leq4−1≤x≤4), so f(5)f(5)f(5) is undefined.

STEP 6

To graph fff, we plot the function for the two parts of the domain separately. For −1≤x<3-1 \leq x <3−1≤x<3, we plot the line y=5xy =5xy=5x. For 3≤x≤43 \leq x \leq43≤x≤4, we plot the line y=x+4y = x +4y=x+4.

STEP 7

STEP 8

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Now, let's evaluate $f(-2), f(0)$ , and $f(5)$ .For $f(-2)$ , we see that $-2$ is not in the domain of $f$ (since $-1 \leq x \leq$ ), so $f(-2)$ is undefined.

For $f(0)$ , we see that $0$ is in the first part of the domain of $f$ (since $-1 \leq x <3$ ). So, we use the first part of the function definition to evaluate $f(0)$ .
$f(0) = \times0 =0$

For $f(5)$ , we see that $5$ is not in the domain of $f$ (since $-1 \leq x \leq4$ ), so $f(5)$ is undefined.

To graph $f$ , we plot the function for the two parts of the domain separately. For $-1 \leq x <3$ , we plot the line $y =5x$ . For $3 \leq x \leq4$ , we plot the line $y = x +4$ .