if $f(x)=(x^{m}+9)^{2}$, which statement about $f(x)$ is true?\n$f(x)$ is an even function for all values of…

if $f(x)=(x^{m}+9)^{2}$, which statement about $f(x)$ is true?\n$f(x)$ is an even function for all values of $m$.\n$f(x)$ is an even function for all even values of $m$.\n$f(x)$ is an odd function for all values of $m$.\n$f(x)$ is an odd function for all odd values of $m$.

if $f(x)=(x^{m}+9)^{2}$, which statement about $f(x)$ is true?\n$f(x)$ is an even function for all values of $m$.\n$f(x)$ is an even function for all even values of $m$.\n$f(x)$ is an odd function for all values of $m$.\n$f(x)$ is an odd function for all odd values of $m$.

Answer

Explanation:

Step1: Recall function - type definitions

An even function satisfies (f(-x)=f(x)) and an odd function satisfies (f(-x)=-f(x)). First, find (f(-x)): [f(-x)=((-x)^{m}+9)^{2}]

Step2: Analyze cases for (m)

Case 1: If (m) is even, then ((-x)^{m}=x^{m}). So (f(-x)=(x^{m}+9)^{2}=f(x)). Case 2: If (m) is odd, then ((-x)^{m}=-x^{m}), and (f(-x)=(-x^{m}+9)^{2}=x^{2m}-18x^{m}+81), while (f(x)=(x^{m}+9)^{2}=x^{2m}+18x^{m}+81). Also, (-f(x)=-x^{2m}-18x^{m}-81). So when (m) is odd, (f(-x)\neq f(x)) and (f(-x)\neq - f(x)).

Answer:

f(x) is an even function for all even values of m.