Answer :
To determine if the function [tex]\( f(x) = 9x^4 - 23 \)[/tex] is even, odd, or neither, we will use the definitions of even and odd functions:
1. Even Function: A function is even if [tex]\( f(x) = f(-x) \)[/tex] for all values of [tex]\( x \)[/tex]. Graphically, this means the function is symmetrical about the y-axis.
2. Odd Function: A function is odd if [tex]\( f(-x) = -f(x) \)[/tex] for all values of [tex]\( x \)[/tex]. Graphically, this means the function is symmetrical about the origin.
Steps to determine the type of function:
1. Original function:
[tex]\[
f(x) = 9x^4 - 23
\][/tex]
2. Substitute [tex]\(-x\)[/tex] for [tex]\(x\)[/tex] in the function to find [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = 9(-x)^4 - 23
\][/tex]
3. Calculate [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = 9x^4 - 23
\][/tex]
Since [tex]\((-x)^4 = x^4\)[/tex], because raising to an even power makes negative disappear.
4. Compare [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex]:
[tex]\[
f(x) = 9x^4 - 23
\][/tex]
[tex]\[
f(-x) = 9x^4 - 23
\][/tex]
Since [tex]\( f(x) = f(-x) \)[/tex], the function is even.
Therefore, the function [tex]\( f(x) = 9x^4 - 23 \)[/tex] is an even function.
1. Even Function: A function is even if [tex]\( f(x) = f(-x) \)[/tex] for all values of [tex]\( x \)[/tex]. Graphically, this means the function is symmetrical about the y-axis.
2. Odd Function: A function is odd if [tex]\( f(-x) = -f(x) \)[/tex] for all values of [tex]\( x \)[/tex]. Graphically, this means the function is symmetrical about the origin.
Steps to determine the type of function:
1. Original function:
[tex]\[
f(x) = 9x^4 - 23
\][/tex]
2. Substitute [tex]\(-x\)[/tex] for [tex]\(x\)[/tex] in the function to find [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = 9(-x)^4 - 23
\][/tex]
3. Calculate [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = 9x^4 - 23
\][/tex]
Since [tex]\((-x)^4 = x^4\)[/tex], because raising to an even power makes negative disappear.
4. Compare [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex]:
[tex]\[
f(x) = 9x^4 - 23
\][/tex]
[tex]\[
f(-x) = 9x^4 - 23
\][/tex]
Since [tex]\( f(x) = f(-x) \)[/tex], the function is even.
Therefore, the function [tex]\( f(x) = 9x^4 - 23 \)[/tex] is an even function.
