Solved on Jan 26, 2024

Describe the transformations of g(x)=xg(x) = |x| to get f(x)=2x3+5f(x) = -2|x-3| + 5. Find the domain and range of f(x)f(x).

STEP 1

Assumptions
1. The base function is g(x)=xg(x) = |x|.
2. The transformed function is f(x)=2x3+5f(x) = -2|x - 3| + 5.
3. We need to describe the transformations applied to g(x)g(x) to get f(x)f(x).
4. We need to find the domain and range of f(x)f(x).

STEP 2

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

STEP 3

Determine the vertical stretch/compression and reflection.
In f(x)f(x), the coefficient of x3|x - 3| is 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 (x3)(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) is +5+5. This indicates a vertical shift upwards by 5 units.

STEP 6

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

STEP 7

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

STEP 8

Determine the range of f(x)f(x).
Since g(x)=xg(x) = |x| has a minimum value of 0 and f(x)f(x) is reflected in the x-axis, the maximum value of f(x)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)g(x) to negative values in f(x)f(x), and the vertical stretch by a factor of 2 doubles all y-values, the range of f(x)f(x) is all values less than or equal to 5.
Range of f(x):(,5]Range\ of\ f(x): (-\infty, 5]

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