Question

Given the function ( f(x) ) below, evaluate ( 3f(-2) - 2f(1) ).

[ f(x) = begin{cases} 7x - 2 & text{if } x leq -3 3x^2 - 2x & text{if } -3 < x < 0 -2sqrt{x} - 1 & text{if } x > 0 end{cases} ]
a) -27
b) -21
c) -15
d) -9

Answer 1

To solve 3f(-2) - 2f(1), f(-2) is calculated using the second part of the function since -2 is between -3 and 0, resulting in 16. f(1) is found using the third part of the function as 1 is greater than 0, resulting in -3. Finally, 3f(-2) - 2f(1) equals 54, which is not an option provided in the question.

Step-by-step explanation:

To evaluate the expression 3f(-2) - 2f(1), we must first determine which part of the piecewise function to use for each value of x. For f(-2), we use the second case of the function because -2 falls between -3 and 0. For f(1), we use the third case since 1 is greater than 0.

First, calculate f(-2) using the second part of the function:

f(-2) = 3(-2)^2 - 2(-2)

f(-2) = 3(4) + 4

f(-2) = 12 + 4

f(-2) = 16

Then, find f(1) using the third part of the function:

f(1) = -2sqrt(1) - 1

f(1) = -2(1) - 1

f(1) = -3

Now, we can evaluate the original expression:

3f(-2) - 2f(1) = 3(16) - 2(-3)

= 48 + 6

= 54

Therefore, the value of 3f(-2) - 2f(1) is 54, which is not listed as one of the provided options.