Solved on Jan 26, 2024

Describe the transformations of $g(x) = |x|$ to get $f(x) = -2|x-3| + 5$ . Find the domain and range of $f(x)$ .

STEP 1

Assumptions
1. The base function is $g(x) = |x|$ .
2. The transformed function is $f(x) = -2|x - 3| + 5$ .
3. We need to describe the transformations applied to $g(x)$ to get $f(x)$ .
4. We need to find the domain and range of $f(x)$ .

STEP 2

Identify the transformations applied to $g(x)$ by comparing it to $f(x)$ .
The general form of transformations for a function $h(x) = a|b(x - h)| + k$ includes:
- Vertical stretch/compression by a factor of $|a|$ . - Reflection in the x-axis if $a$ is negative. - Horizontal stretch/compression by a factor of $|b|$ . - Horizontal shift by $h$ units. - Vertical shift by $k$ units.

STEP 3

Determine the vertical stretch/compression and reflection.
In $f(x)$ , the coefficient of $|x - 3|$ is $-2$ . This indicates a vertical stretch by a factor of 2 and a reflection in the x-axis because the coefficient is negative.

STEP 4

Determine the horizontal shift.
The expression inside the absolute value is $(x - 3)$ . This indicates a horizontal shift to the right by 3 units.

STEP 5

Determine the vertical shift.
The constant term outside the absolute value in $f(x)$ is $+5$ . This indicates a vertical shift upwards by 5 units.

STEP 6

Combine the transformations.
The function $f(x)$ is obtained from $g(x)$ by:
1. Reflecting $g(x)$ in the x-axis.
2. Stretching $g(x)$ vertically by a factor of 2.
3. Shifting $g(x)$ horizontally to the right by 3 units.
4. Shifting $g(x)$ vertically upwards by 5 units.

STEP 7

Determine the domain of $f(x)$ .
The domain of the base function $g(x) = |x|$ is all real numbers, and none of the transformations change the domain. Therefore, the domain of $f(x)$ is also all real numbers.
$Domain\ of\ f(x): (-\infty, \infty)$

STEP 8

Determine the range of $f(x)$ .
Since $g(x) = |x|$ has a minimum value of 0 and $f(x)$ is reflected in the x-axis, the maximum value of $f(x)$ without the vertical shift would be 0. After applying the vertical stretch by a factor of 2, the maximum value would be 0. Finally, the vertical shift upwards by 5 units moves the maximum value to 5. Since the reflection in the x-axis inverts all positive values of $g(x)$ to negative values in $f(x)$ , and the vertical stretch by a factor of 2 doubles all y-values, the range of $f(x)$ is all values less than or equal to 5.
$Range\ of\ f(x): (-\infty, 5]$

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Describe the transformations of $g(x) = |x|$ to get $f(x) = -2|x-3| + 5$ . Find the domain and range of $f(x)$ .

STEP 1

Assumptions
1. The base function is $g(x) = |x|$ .
2. The transformed function is $f(x) = -2|x - 3| + 5$ .
3. We need to describe the transformations applied to $g(x)$ to get $f(x)$ .
4. We need to find the domain and range of $f(x)$ .

STEP 2

STEP 3

Determine the vertical stretch/compression and reflection.
In $f(x)$ , the coefficient of $|x - 3|$ is $-2$ . This indicates a vertical stretch by a factor of 2 and a reflection in the x-axis because the coefficient is negative.

STEP 4

Determine the horizontal shift.
The expression inside the absolute value is $(x - 3)$ . This indicates a horizontal shift to the right by 3 units.

STEP 5

Determine the vertical shift.
The constant term outside the absolute value in $f(x)$ is $+5$ . This indicates a vertical shift upwards by 5 units.

STEP 6

Combine the transformations.
The function $f(x)$ is obtained from $g(x)$ by:
1. Reflecting $g(x)$ in the x-axis.
2. Stretching $g(x)$ vertically by a factor of 2.
3. Shifting $g(x)$ horizontally to the right by 3 units.
4. Shifting $g(x)$ vertically upwards by 5 units.

STEP 7

Determine the domain of $f(x)$ .
The domain of the base function $g(x) = |x|$ is all real numbers, and none of the transformations change the domain. Therefore, the domain of $f(x)$ is also all real numbers.
$Domain\ of\ f(x): (-\infty, \infty)$

STEP 8

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Describe the transformations of g(x)=∣x∣g(x) = |x|g(x)=∣x∣ to get f(x)=−2∣x−3∣+5f(x) = -2|x-3| + 5f(x)=−2∣x−3∣+5. Find the domain and range of f(x)f(x)f(x).

STEP 1

STEP 2

STEP 3

Determine the vertical stretch/compression and reflection.In f(x)f(x)f(x), the coefficient of ∣x−3∣|x - 3|∣x−3∣ is −2-2−2. This indicates a vertical stretch by a factor of 2 and a reflection in the x-axis because the coefficient is negative.

STEP 4

Determine the horizontal shift.The expression inside the absolute value is (x−3)(x - 3)(x−3). This indicates a horizontal shift to the right by 3 units.

STEP 5

Determine the vertical shift.The constant term outside the absolute value in f(x)f(x)f(x) is +5+5+5. This indicates a vertical shift upwards by 5 units.

STEP 6

STEP 7

STEP 8

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

Describe the transformations of $g(x) = |x|$ to get $f(x) = -2|x-3| + 5$ . Find the domain and range of $f(x)$ .

Determine the vertical stretch/compression and reflection.
In $f(x)$ , the coefficient of $|x - 3|$ is $-2$ . This indicates a vertical stretch by a factor of 2 and a reflection in the x-axis because the coefficient is negative.

Determine the horizontal shift.
The expression inside the absolute value is $(x - 3)$ . This indicates a horizontal shift to the right by 3 units.

Determine the vertical shift.
The constant term outside the absolute value in $f(x)$ is $+5$ . This indicates a vertical shift upwards by 5 units.