Define the piecewise function:

f(x) = {
x² - x + 10, if x ≤ 3
x - 5, if x ≥ 3
}

Find:
a. f(2)
b. f(-2)
c. f(10)
d. f(7)
e. f(3)

The evaluations for the piecewise function yield the following:

f ( 2 ) = 12 , f ( − 2 ) = 16 , f ( 10 ) = 5 , f ( 7 ) = 2 , and f ( 3 ) = − 2 .

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Answered by Anonymous | 2025-07-04

Let's evaluate the piecewise function for the given values of x using the appropriate part of the function, based on the value of x .
The piecewise function is defined as:
f ( x ) = { x 2 − x + 10 , x − 5 , ​ if x ≤ 3 if x ≥ 3 ​
Let's find the value of f ( x ) for different values of x :
(a) f ( 2 ) :

Since 2 ≤ 3 , use the first part of the function: f ( 2 ) = 2 2 − 2 + 10 .
Calculate: f ( 2 ) = 4 − 2 + 10 = 12 .

(b) f ( − 2 ) :

Since − 2 ≤ 3 , use the first part of the function: f ( − 2 ) = ( − 2 ) 2 − ( − 2 ) + 10 .
Calculate: f ( − 2 ) = 4 + 2 + 10 = 16 .

(c) f ( 10 ) :

Since 10 ≥ 3 , use the second part of the function: f ( 10 ) = 10 − 5 .
Calculate: f ( 10 ) = 5 .

(d) f ( 7 ) :

Since 7 ≥ 3 , use the second part of the function: f ( 7 ) = 7 − 5 .
Calculate: f ( 7 ) = 2 .

(e) f ( 3 ) :

Since 3 ≤ 3 , we use the first part of the function: f ( 3 ) = 3 2 − 3 + 10 .
Calculate: f ( 3 ) = 9 − 3 + 10 = 16 .

Thus, the evaluated values are:

f ( 2 ) = 12
f ( − 2 ) = 16
f ( 10 ) = 5
f ( 7 ) = 2
f ( 3 ) = 16

Answered by DanielJosephParker | 2025-07-06