To determine which of the given functions is an even function, let’s review the definition and then analyze each function individually.
Definition: Even Function
A function [tex]\( f(x) \)[/tex] is called even if for all [tex]\( x \)[/tex] in its domain, [tex]\( f(x) = f(-x) \)[/tex].
Let us analyze each function to determine if it satisfies this condition.
1. [tex]\( f(x) = |x| \)[/tex]:
- [tex]\( f(x) = |x| \)[/tex]
- [tex]\( f(-x) = |-x| = |x| \)[/tex]
- Since [tex]\( f(x) = f(-x) \)[/tex], the function [tex]\( f(x) = |x| \)[/tex] is even.
2. [tex]\( f(x) = x^3 - 1 \)[/tex]:
- [tex]\( f(x) = x^3 - 1 \)[/tex]
- [tex]\( f(-x) = (-x)^3 - 1 = -x^3 - 1 \)[/tex]
- Since [tex]\( f(x) \neq f(-x) \)[/tex], the function [tex]\( f(x) = x^3 - 1 \)[/tex] is not even.
3. [tex]\( f(x) = -3x \)[/tex]:
- [tex]\( f(x) = -3x \)[/tex]
- [tex]\( f(-x) = -3(-x) = 3x \)[/tex]
- Since [tex]\( f(x) \neq f(-x) \)[/tex], the function [tex]\( f(x) = -3x \)[/tex] is not even.
4. [tex]\( f(x) = \sqrt[3]{x} \)[/tex]:
- [tex]\( f(x) = \sqrt[3]{x} \)[/tex]
- [tex]\( f(-x) = \sqrt[3]{-x} \)[/tex]
- Since [tex]\( f(x) \neq f(-x) \)[/tex], the function [tex]\( f(x) = \sqrt[3]{x} \)[/tex] is not even.
Conclusion:
Among the given functions, only [tex]\( f(x) = |x| \)[/tex] satisfies the condition [tex]\( f(x) = f(-x) \)[/tex], which means [tex]\( f(x) = |x| \)[/tex] is an even function.
Thus, the even function is:
[tex]\[ f(x) = |x| \][/tex]
