To determine which of the given functions is an even function, we need to understand the definition of an even function. A function [tex]\( f(x) \)[/tex] is considered even if it satisfies the property [tex]\( f(x) = f(-x) \)[/tex] for all [tex]\( x \)[/tex] in its domain.
Let's examine each function step by step:
1. Function [tex]\( f(x) = |x| \)[/tex]
- For [tex]\( x \geq 0 \)[/tex], [tex]\( |x| = x \)[/tex].
- For [tex]\( x < 0 \)[/tex], [tex]\( |x| = -x \)[/tex].
- Evaluate [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex]:
[tex]\[
f(x) = |x|
\][/tex]
[tex]\[
f(-x) = |-x| = x
\][/tex]
- Since [tex]\( f(x) = f(-x) \)[/tex] holds true for all [tex]\( x \)[/tex], the function [tex]\( f(x) = |x| \)[/tex] is even.
2. Function [tex]\( f(x) = x^3 - 1 \)[/tex]
- Evaluate [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex]:
[tex]\[
f(x) = x^3 - 1
\][/tex]
[tex]\[
f(-x) = (-x)^3 - 1 = -x^3 - 1
\][/tex]
- Compare [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex]:
[tex]\[
x^3 - 1 \neq -x^3 - 1
\][/tex]
- Since [tex]\( f(x) \neq f(-x) \)[/tex], the function [tex]\( f(x) = x^3 - 1 \)[/tex] is not even.
3. Function [tex]\( f(x) = -3x \)[/tex]
- Evaluate [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex]:
[tex]\[
f(x) = -3x
\][/tex]
[tex]\[
f(-x) = -3(-x) = 3x
\][/tex]
- Compare [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex]:
[tex]\[
-3x \neq 3x
\][/tex]
- Since [tex]\( f(x) \neq f(-x) \)[/tex], the function [tex]\( f(x) = -3x \)[/tex] is not even.
4. Function [tex]\( f(x) = \sqrt[3]{x} \)[/tex]
- Evaluate [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex]:
[tex]\[
f(x) = \sqrt[3]{x}
\][/tex]
[tex]\[
f(-x) = \sqrt[3]{-x} = -\sqrt[3]{x}
\][/tex]
- Compare [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex]:
[tex]\[
\sqrt[3]{x} \neq -\sqrt[3]{x}
\][/tex]
- Since [tex]\( f(x) \neq f(-x) \)[/tex], the function [tex]\( f(x) = \sqrt[3]{x} \)[/tex] is not even.
In conclusion, out of the given functions, the only function that is even is:
[tex]\[ f(x) = |x| \][/tex]
