Question

Find the values of f(x):

f(x) = {
x² if -1 ≤ x ≤ 0
8 - 9x if 0 ≤ x ≤ 2
x⁴ if x ≥ 2
}

Answer 1

The values of f(x) are as follows:

- For -1 ≤ x ≤ 0, f(x) = x².

- For 0 ≤ x ≤ 2, f(x) = 8 - 9x.

- For x ≥ 2, f(x) = x⁴.

Explanation:

The function f(x) is defined piecewise for different intervals of x. For x values between -1 and 0 inclusive, the function takes the form f(x) = x². In the range from 0 to 2 inclusive, the function is represented by f(x) = 8 - 9x. Lastly, for x values greater than or equal to 2, the function is given by f(x) = x⁴.

Within the range -1 to 0, the output of the function f(x) is determined by the expression x², which yields values for f(x) as x varies from -1 to 0. In the interval between 0 and 2, the function f(x) is defined as 8 - 9x, meaning it produces outputs based on the expression 8 - 9x for x values from 0 to 2. When x is greater than or equal to 2, the function f(x) takes the form x⁴, resulting in values of f(x) calculated using x raised to the power of 4 for x values that are 2 or larger. These defined ranges provide distinct representations of f(x) based on the specified conditions for x.

Answer 2

The given function f(x) can be expressed in three different ranges of x: -1 ≤ x ≤ 0, 0 ≤ x ≤ 2, and x ≥ 2. In each range, the expression for f(x) is different. For example, if x is between -1 and 0, f(x) equals x squared. If x is between 0 and 2, f(x) equals -9 times x. If x is greater than or equal to 2, f(x) equals 2 times x raised to the power of 4.

Step-by-step explanation:

The given function can be written as:

f(x) = x², if -1 ≤ x ≤ 0

f(x) = -9x, if 0 ≤ x ≤ 2

f(x) = 2x⁴, if x ≥ 2

To find the values of f(x) for different values of x, we need to consider the different ranges of x.

If -1 ≤ x ≤ 0, f(x) = x². For example, if x = -0.5, f(-0.5) = (-0.5)² = 0.25.

If 0 ≤ x ≤ 2, f(x) = -9x. For example, if x = 1.5, f(1.5) = -9(1.5) = -13.5.

If x ≥ 2, f(x) = 2x⁴. For example, if x = 3, f(3) = 2(3)⁴ = 162.