To determine which of the functions are even, we need to check if each function satisfies the definition of an even function. A function [tex]\( f(x) \)[/tex] is called even if [tex]\( f(x) = f(-x) \)[/tex] for all [tex]\( x \)[/tex] in its domain.
Let's test each given function accordingly:
1. Function 1: [tex]\( f(x) = (x-1)^2 \)[/tex]
[tex]\[
f(x) = (x-1)^2
\][/tex]
To test if it is even, we need to check if [tex]\( f(x) = f(-x) \)[/tex]:
[tex]\[
f(-x) = (-x-1)^2 = (-(x+1))^2 = (x+1)^2 \neq (x-1)^2
\][/tex]
Hence, [tex]\( f(x) = (x-1)^2 \)[/tex] is not an even function.
2. Function 2: [tex]\( f(x) = 8x \)[/tex]
[tex]\[
f(x) = 8x
\][/tex]
To test if it is even, we need to check if [tex]\( f(x) = f(-x) \)[/tex]:
[tex]\[
f(-x) = 8(-x) = -8x \neq 8x
\][/tex]
Hence, [tex]\( f(x) = 8x \)[/tex] is not an even function.
3. Function 3: [tex]\( f(x) = x^2 - x \)[/tex]
[tex]\[
f(x) = x^2 - x
\][/tex]
To test if it is even, we need to check if [tex]\( f(x) = f(-x) \)[/tex]:
[tex]\[
f(-x) = (-x)^2 - (-x) = x^2 + x \neq x^2 - x
\][/tex]
Hence, [tex]\( f(x) = x^2 - x \)[/tex] is not an even function.
4. Function 4: [tex]\( f(x) = 7 \)[/tex]
[tex]\[
f(x) = 7
\][/tex]
To test if it is even, we need to check if [tex]\( f(x) = f(-x) \)[/tex]:
[tex]\[
f(-x) = 7
\][/tex]
Clearly, [tex]\( f(x) = 7 = f(-x) \)[/tex] for all [tex]\( x \)[/tex].
Hence, the only even function from the given list is:
[tex]\[ f(x) = 7 \][/tex]
