Answer:
The odd function among the given choices are:
Step-by-step explanation:
We have to check which of the following function is odd.
We know that a function f(x) is said to be an odd function if:
[tex]f(-x)= -f(x)[/tex]
1)
[tex]f(x)=x^3-x^2[/tex]
on replacing x with -x we have:
[tex]f(-x)=(-x)^3-(-x)^2\\\\f(-x)=-x^3-x^2\neq -f(x)[/tex]
Hence, the given function f(x) is not an odd function.
2)
[tex]f(x)=x^5-3x^3+2x[/tex]
on replacing x with -x we have:
[tex]f(-x)=(-x)^5-3(-x)^3+2(-x)\\\\f(-x)=-x^5+3x^3-2x\\\\f(-x)=-(x^5-3x^3+2x)\\\\f(-x)=-f(x)[/tex]
Hence, function f(x) is an odd function.
3)
[tex]f(x)=4x+9[/tex]
on replacing x with -x we have:
[tex]f(-x)=4(-x)+9\\\\f(-x)=-4x+9\\\\f(-x)\neq -f(x)[/tex]
Hence, function f(x) is not an odd function.
4)
[tex]f(x)=\dfrac{1}{x}[/tex]
on replacing x with -x we have:
[tex]f(-x)=\dfrac{1}{(-x)}\\\\f(-x)=\dfrac{-1}{x}\\\\f(-x)=-f(x)[/tex]
Hence, the given function f(x) is an odd function.
A function is odd if and only if:
f(-x) = -f(x)
For every value of x in its domain.
We will see that the odd functions are:
f(x) = x^5 - 3*x^3 + 2*x
f(x) = 1/x
Now we only need to test the above condition for all the given functions:
1) f(x) = x^3 - x^2
Evaluating this in -x we get:
f(-x) = (-x)^3 - (-x)^2 = -x^3 - x^2
Here we can see that:
f(-x) ≠ -f(x)
Thus, this function is not odd.
2) f(x) = x^5 - 3*x^3 + 2*x
Evaluating in -x we get:
f(-x) = (-x)^5 - 3*(-x)^3 + 2*(-x)
= -x^5 + 3*x^3 - 2*x = -f(x)
This function is odd.
3) f(x) = 4x + 9
Evaluating in -x we get:
f(-x) = 4*(-x) + 9 = -4*x + 9
while:
-f(x) = -( 4*x + 9) = -4*x - 9
Thus: f(-x) ≠ -f(x)
Then this function is not odd.
4) f(x) = 1/x
Evaluating in -x we get:
f(-x) = 1/(-x) = -1/x
Then we can see that:
f(-x) = -f(x)
Thus this function is odd.
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