Answer :
To find [tex]\(-h(x)\)[/tex], we first need to establish what [tex]\(h(x)\)[/tex] is, based on the provided information about [tex]\(h(-x)\)[/tex].
Given:
[tex]\[ h(-x) = -9x^9 - 4x^3 \][/tex]
We want to find the expression for [tex]\(-h(x)\)[/tex]. To do this, we need to determine [tex]\(h(x)\)[/tex].
1. Identify [tex]\(h(x)\)[/tex]:
Since [tex]\(h(-x)\)[/tex] is provided, and applies to all [tex]\(-x\)[/tex], we replace [tex]\(-x\)[/tex] with [tex]\(x\)[/tex] to infer a possible expression for [tex]\(h(x)\)[/tex]:
[tex]\[
h(x) = 9x^9 + 4x^3
\][/tex]
(This step assumes that the effect of plugging [tex]\(-x\)[/tex] into [tex]\(h\)[/tex] results in flipping the signs appropriately according to even and odd powers of [tex]\(x\)[/tex]. In this case, [tex]\(x^9\)[/tex] is odd and [tex]\(x^3\)[/tex] is odd, so their signs flip entirely when [tex]\(-x\)[/tex] is plugged into the function.)
2. Compute [tex]\(-h(x)\)[/tex]:
[tex]\[
-h(x) = -(9x^9 + 4x^3)
\][/tex]
When you distribute the negative sign, you get:
[tex]\[
-h(x) = -9x^9 - 4x^3
\][/tex]
This is the expression for [tex]\(-h(x)\)[/tex].
Given:
[tex]\[ h(-x) = -9x^9 - 4x^3 \][/tex]
We want to find the expression for [tex]\(-h(x)\)[/tex]. To do this, we need to determine [tex]\(h(x)\)[/tex].
1. Identify [tex]\(h(x)\)[/tex]:
Since [tex]\(h(-x)\)[/tex] is provided, and applies to all [tex]\(-x\)[/tex], we replace [tex]\(-x\)[/tex] with [tex]\(x\)[/tex] to infer a possible expression for [tex]\(h(x)\)[/tex]:
[tex]\[
h(x) = 9x^9 + 4x^3
\][/tex]
(This step assumes that the effect of plugging [tex]\(-x\)[/tex] into [tex]\(h\)[/tex] results in flipping the signs appropriately according to even and odd powers of [tex]\(x\)[/tex]. In this case, [tex]\(x^9\)[/tex] is odd and [tex]\(x^3\)[/tex] is odd, so their signs flip entirely when [tex]\(-x\)[/tex] is plugged into the function.)
2. Compute [tex]\(-h(x)\)[/tex]:
[tex]\[
-h(x) = -(9x^9 + 4x^3)
\][/tex]
When you distribute the negative sign, you get:
[tex]\[
-h(x) = -9x^9 - 4x^3
\][/tex]
This is the expression for [tex]\(-h(x)\)[/tex].
