Question Rewrite the piecewise equation $y = \{\begin{array}{ll} \frac{1}{5}x + 8 & \text{if } x \leq -5\\ -x - 1 & \text{if } -4 \leq x < -2\\ -\frac{5}{4}x + 2 & \text{if } 0 \leq x \end{array}$ . Find the domain and evaluate $f(a)$ for $a = -1, -4, -9, 8$ .

Studdy Solution

STEP 1

Assumptions
1. The given function is a piecewise function with three different expressions for different intervals of $x$ .
2. The domain of a function is the set of all possible input values (values of $x$ ) which allow the function to work and will output real numbers.

STEP 2

To find the domain of $f(x)$ , we need to look at the intervals given in the piecewise function and combine them.

STEP 3

The first interval is $x \leq -5$ . This means all $x$ values less than or equal to $-5$ are included in the domain.

STEP 4

The second interval is $-4 \leq x < -2$ . This means all $x$ values from $-4$ inclusive to $-2$ exclusive are included in the domain.

STEP 5

The third interval is $0 \leq x$ . This means all $x$ values greater than or equal to $0$ are included in the domain.

STEP 6

Combine all intervals to find the domain. The domain is all $x$ values that are less than or equal to $-5$ , between $-4$ and $-2$ (including $-4$ but not $-2$ ), and greater than or equal to $0$ .

STEP 7

Write the domain of $f(x)$ using interval notation.
$Domain\ of\ f(x) = (-\infty, -5] \cup [-4, -2) \cup [0, \infty)$

STEP 8

Now, we will evaluate $f(a)$ for each given value of $a$ .

STEP 9

To find $f(-1)$ , we look at the intervals given in the piecewise function to determine which expression to use. Since $-1$ is not included in any of the intervals, $f(-1)$ is undefined.
$f(-1) = \text{undefined}$

STEP 10

To find $f(-4)$ , we use the second expression because $-4$ is in the interval $-4 \leq x < -2$ .
$f(-4) = -1(-4) - 1$

STEP 11

Calculate $f(-4)$ .
$f(-4) = 4 - 1 = 3$

STEP 12

To find $f(-9)$ , we use the first expression because $-9$ is in the interval $x \leq -5$ .
$f(-9) = \frac{1}{5}(-9) + 8$

STEP 13

Calculate $f(-9)$ .
$f(-9) = -\frac{9}{5} + 8$

STEP 14

Convert $8$ to a fraction with a denominator of $5$ to combine with $-\frac{9}{5}$ .
$8 = \frac{40}{5}$

STEP 15

Calculate $f(-9)$ .
$f(-9) = -\frac{9}{5} + \frac{40}{5} = \frac{31}{5} = 6.2$

STEP 16

To find $f(8)$ , we use the third expression because $8$ is in the interval $0 \leq x$ .
$f(8) = -\frac{5}{4}(8) + 2$

STEP 17

Calculate $f(8)$ .
$f(8) = -10 + 2 = -8$

STEP 18

Summarize the evaluations of $f(a)$ for each given value of $a$ .
1. $f(-1) = \text{undefined}$
2. $f(-4) = 3$
3. $f(-9) = 6.2$
4. $f(8) = -8$

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Studdy Solution

STEP 1

Assumptions1. The given function is a piecewise function with three different expressions for different intervals of xxx.2. The domain of a function is the set of all possible input values (values of xxx) which allow the function to work and will output real numbers.

STEP 2

To find the domain of f(x)f(x)f(x), we need to look at the intervals given in the piecewise function and combine them.

STEP 3

The first interval is x≤−5x \leq -5x≤−5. This means all xxx values less than or equal to −5-5−5 are included in the domain.

STEP 4

The second interval is −4≤x<−2-4 \leq x < -2−4≤x<−2. This means all xxx values from −4-4−4 inclusive to −2-2−2 exclusive are included in the domain.

STEP 5

The third interval is 0≤x0 \leq x0≤x. This means all xxx values greater than or equal to 000 are included in the domain.

STEP 6

Combine all intervals to find the domain. The domain is all xxx values that are less than or equal to −5-5−5, between −4-4−4 and −2-2−2 (including −4-4−4 but not −2-2−2), and greater than or equal to 000.

STEP 7

Write the domain of f(x)f(x)f(x) using interval notation.Domain of f(x)=(−∞,−5]∪[−4,−2)∪[0,∞)Domain\ of\ f(x) = (-\infty, -5] \cup [-4, -2) \cup [0, \infty)Domain of f(x)=(−∞,−5]∪[−4,−2)∪[0,∞)

STEP 8

Now, we will evaluate f(a)f(a)f(a) for each given value of aaa.

STEP 9

To find f(−1)f(-1)f(−1), we look at the intervals given in the piecewise function to determine which expression to use. Since −1-1−1 is not included in any of the intervals, f(−1)f(-1)f(−1) is undefined.f(−1)=undefinedf(-1) = \text{undefined}f(−1)=undefined

STEP 10

To find f(−4)f(-4)f(−4), we use the second expression because −4-4−4 is in the interval −4≤x<−2-4 \leq x < -2−4≤x<−2.f(−4)=−1(−4)−1f(-4) = -1(-4) - 1f(−4)=−1(−4)−1

STEP 11

Calculate f(−4)f(-4)f(−4).f(−4)=4−1=3f(-4) = 4 - 1 = 3f(−4)=4−1=3

STEP 12

To find f(−9)f(-9)f(−9), we use the first expression because −9-9−9 is in the interval x≤−5x \leq -5x≤−5.f(−9)=15(−9)+8f(-9) = \frac{1}{5}(-9) + 8f(−9)=51​(−9)+8

STEP 13

Calculate f(−9)f(-9)f(−9).f(−9)=−95+8f(-9) = -\frac{9}{5} + 8f(−9)=−59​+8

STEP 14

Convert 888 to a fraction with a denominator of 555 to combine with −95-\frac{9}{5}−59​.8=4058 = \frac{40}{5}8=540​

STEP 15

Calculate f(−9)f(-9)f(−9).f(−9)=−95+405=315=6.2f(-9) = -\frac{9}{5} + \frac{40}{5} = \frac{31}{5} = 6.2f(−9)=−59​+540​=531​=6.2

STEP 16

To find f(8)f(8)f(8), we use the third expression because 888 is in the interval 0≤x0 \leq x0≤x.f(8)=−54(8)+2f(8) = -\frac{5}{4}(8) + 2f(8)=−45​(8)+2

STEP 17

Calculate f(8)f(8)f(8).f(8)=−10+2=−8f(8) = -10 + 2 = -8f(8)=−10+2=−8

STEP 18

Summarize the evaluations of f(a)f(a)f(a) for each given value of aaa.1. f(−1)=undefinedf(-1) = \text{undefined}f(−1)=undefined2. f(−4)=3f(-4) = 3f(−4)=33. f(−9)=6.2f(-9) = 6.2f(−9)=6.24. f(8)=−8f(8) = -8f(8)=−8

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Assumptions
1. The given function is a piecewise function with three different expressions for different intervals of $x$ .
2. The domain of a function is the set of all possible input values (values of $x$ ) which allow the function to work and will output real numbers.

To find the domain of $f(x)$ , we need to look at the intervals given in the piecewise function and combine them.

The first interval is $x \leq -5$ . This means all $x$ values less than or equal to $-5$ are included in the domain.

The second interval is $-4 \leq x < -2$ . This means all $x$ values from $-4$ inclusive to $-2$ exclusive are included in the domain.

The third interval is $0 \leq x$ . This means all $x$ values greater than or equal to $0$ are included in the domain.

Combine all intervals to find the domain. The domain is all $x$ values that are less than or equal to $-5$ , between $-4$ and $-2$ (including $-4$ but not $-2$ ), and greater than or equal to $0$ .

Write the domain of $f(x)$ using interval notation.
$Domain\ of\ f(x) = (-\infty, -5] \cup [-4, -2) \cup [0, \infty)$

Now, we will evaluate $f(a)$ for each given value of $a$ .

To find $f(-1)$ , we look at the intervals given in the piecewise function to determine which expression to use. Since $-1$ is not included in any of the intervals, $f(-1)$ is undefined.
$f(-1) = \text{undefined}$

To find $f(-4)$ , we use the second expression because $-4$ is in the interval $-4 \leq x < -2$ .
$f(-4) = -1(-4) - 1$

Calculate $f(-4)$ .
$f(-4) = 4 - 1 = 3$

To find $f(-9)$ , we use the first expression because $-9$ is in the interval $x \leq -5$ .
$f(-9) = \frac{1}{5}(-9) + 8$

Calculate $f(-9)$ .
$f(-9) = -\frac{9}{5} + 8$

Convert $8$ to a fraction with a denominator of $5$ to combine with $-\frac{9}{5}$ .
$8 = \frac{40}{5}$

Calculate $f(-9)$ .
$f(-9) = -\frac{9}{5} + \frac{40}{5} = \frac{31}{5} = 6.2$

To find $f(8)$ , we use the third expression because $8$ is in the interval $0 \leq x$ .
$f(8) = -\frac{5}{4}(8) + 2$

Calculate $f(8)$ .
$f(8) = -10 + 2 = -8$

Summarize the evaluations of $f(a)$ for each given value of $a$ .
1. $f(-1) = \text{undefined}$
2. $f(-4) = 3$
3. $f(-9) = 6.2$
4. $f(8) = -8$