To determine which of the given functions are even, we need to check if each function satisfies the property of even functions: [tex]\( f(x) = f(-x) \)[/tex]. Let's evaluate each function step-by-step.
1. Function: [tex]\( f(x) = |x| \)[/tex]
- Evaluate [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = |-x| = |x|
\][/tex]
- Since [tex]\( f(x) = f(-x) \)[/tex], [tex]\( f(x) = |x| \)[/tex] is an even function.
2. Function: [tex]\( f(x) = x^3 - 1 \)[/tex]
- Evaluate [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = (-x)^3 - 1 = -x^3 - 1
\][/tex]
- Compare [tex]\( f(-x) \)[/tex] and [tex]\( f(x) \)[/tex]:
[tex]\[
f(-x) = -x^3 - 1 \quad \text{and} \quad f(x) = x^3 - 1
\][/tex]
- Since [tex]\( f(x) \neq f(-x) \)[/tex], [tex]\( f(x) = x^3 - 1 \)[/tex] is not an even function.
3. Function: [tex]\( f(x) = -3x \)[/tex]
- Evaluate [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = -3(-x) = 3x
\][/tex]
- Compare [tex]\( f(-x) \)[/tex] and [tex]\( f(x) \)[/tex]:
[tex]\[
f(-x) = 3x \quad \text{and} \quad f(x) = -3x
\][/tex]
- Since [tex]\( f(x) \neq f(-x) \)[/tex], [tex]\( f(x) = -3x \)[/tex] is not an even function.
4. Function: [tex]\( f(x) = \sqrt[3]{x} \)[/tex]
- Evaluate [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = \sqrt[3]{-x} = -\sqrt[3]{x}
\][/tex]
- Compare [tex]\( f(-x) \)[/tex] and [tex]\( f(x) \)[/tex]:
[tex]\[
f(-x) = -\sqrt[3]{x} \quad \text{and} \quad f(x) = \sqrt[3]{x}
\][/tex]
- Since [tex]\( f(x) \neq f(-x) \)[/tex], [tex]\( f(x) = \sqrt[3]{x} \)[/tex] is not an even function.
From our analysis, only:
[tex]\[ f(x) = |x| \][/tex]
is an even function.
