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Answer :
To determine which of the given statements is true for the function [tex]\( f(x) = x^{2/3} \)[/tex], we need to analyze its properties.
### Checking if [tex]\( f(x) \)[/tex] is an even function:
A function [tex]\( f(x) \)[/tex] is even if [tex]\( f(-x) = f(x) \)[/tex] for all [tex]\( x \)[/tex] in the domain of [tex]\( f(x) \)[/tex].
Let's compute [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = (-x)^{2/3} \][/tex]
Since the exponent [tex]\(\frac{2}{3}\)[/tex] is even (because 2 is even), we have:
[tex]\[ (-x)^{2/3} = (x^2)^{1/3} = x^{2/3} \][/tex]
So,
[tex]\[ f(-x) = f(x) \][/tex]
This shows that [tex]\( f(x) = x^{2/3} \)[/tex] is an even function.
### Checking if [tex]\( f(x) \)[/tex] is an odd function:
A function [tex]\( f(x) \)[/tex] is odd if [tex]\( f(-x) = -f(x) \)[/tex] for all [tex]\( x \)[/tex] in the domain of [tex]\( f(x) \)[/tex].
Let's check this:
[tex]\[ f(-x) = (-x)^{2/3} = x^{2/3} \][/tex]
[tex]\[ -f(x) = -x^{2/3} \][/tex]
Clearly,
[tex]\[ f(-x) = x^{2/3} \neq -x^{2/3} \][/tex]
So, [tex]\( f(-x) \neq -f(x) \)[/tex]
This shows that [tex]\( f(x) = x^{2/3} \)[/tex] is not an odd function.
### Symmetry considerations:
1. Symmetric with respect to the [tex]\( x \)[/tex]-axis:
For symmetry with respect to the [tex]\( x \)[/tex]-axis, whenever [tex]\((x, y)\)[/tex] is on the graph, [tex]\((x, -y)\)[/tex] must also be on the graph. This would mean that for some value of [tex]\( x \)[/tex], [tex]\( f(x) = -f(x) \)[/tex].
Since [tex]\( f(x) = x^{2/3} \)[/tex] can't be negative for real [tex]\( x \)[/tex], it doesn't satisfy this property. Hence, the graph is not symmetric with respect to the [tex]\( x \)[/tex]-axis.
2. Symmetric with respect to the origin:
For symmetry with respect to the origin, [tex]\( f(-x) = -f(x) \)[/tex]. As we've established, [tex]\( f(-x) = f(x) \neq -f(x) \)[/tex], so the graph is not symmetric with respect to the origin.
Combining these results, the only true statement is:
[tex]\[ \text{(B) } f \text{ is an even function.} \][/tex]
Therefore, the correct answer is:
(B) [tex]\( f \)[/tex] is an even function.
### Checking if [tex]\( f(x) \)[/tex] is an even function:
A function [tex]\( f(x) \)[/tex] is even if [tex]\( f(-x) = f(x) \)[/tex] for all [tex]\( x \)[/tex] in the domain of [tex]\( f(x) \)[/tex].
Let's compute [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = (-x)^{2/3} \][/tex]
Since the exponent [tex]\(\frac{2}{3}\)[/tex] is even (because 2 is even), we have:
[tex]\[ (-x)^{2/3} = (x^2)^{1/3} = x^{2/3} \][/tex]
So,
[tex]\[ f(-x) = f(x) \][/tex]
This shows that [tex]\( f(x) = x^{2/3} \)[/tex] is an even function.
### Checking if [tex]\( f(x) \)[/tex] is an odd function:
A function [tex]\( f(x) \)[/tex] is odd if [tex]\( f(-x) = -f(x) \)[/tex] for all [tex]\( x \)[/tex] in the domain of [tex]\( f(x) \)[/tex].
Let's check this:
[tex]\[ f(-x) = (-x)^{2/3} = x^{2/3} \][/tex]
[tex]\[ -f(x) = -x^{2/3} \][/tex]
Clearly,
[tex]\[ f(-x) = x^{2/3} \neq -x^{2/3} \][/tex]
So, [tex]\( f(-x) \neq -f(x) \)[/tex]
This shows that [tex]\( f(x) = x^{2/3} \)[/tex] is not an odd function.
### Symmetry considerations:
1. Symmetric with respect to the [tex]\( x \)[/tex]-axis:
For symmetry with respect to the [tex]\( x \)[/tex]-axis, whenever [tex]\((x, y)\)[/tex] is on the graph, [tex]\((x, -y)\)[/tex] must also be on the graph. This would mean that for some value of [tex]\( x \)[/tex], [tex]\( f(x) = -f(x) \)[/tex].
Since [tex]\( f(x) = x^{2/3} \)[/tex] can't be negative for real [tex]\( x \)[/tex], it doesn't satisfy this property. Hence, the graph is not symmetric with respect to the [tex]\( x \)[/tex]-axis.
2. Symmetric with respect to the origin:
For symmetry with respect to the origin, [tex]\( f(-x) = -f(x) \)[/tex]. As we've established, [tex]\( f(-x) = f(x) \neq -f(x) \)[/tex], so the graph is not symmetric with respect to the origin.
Combining these results, the only true statement is:
[tex]\[ \text{(B) } f \text{ is an even function.} \][/tex]
Therefore, the correct answer is:
(B) [tex]\( f \)[/tex] is an even function.
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