Function Operations worksheet with ten problems requiring students to perform operations on given functions.

Problem: Function Operations

The task involves performing various operations on functions, including composition of functions and division of functions. Let's solve each problem step by step.

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#### 1) Given:
- \( f(n) = -10n - 5 \)
- \( h(n) = 5n^2 - 9 \)
- Find \( (f \cdot h)(n) \)

Solution:
The operation \( (f \cdot h)(n) \) means we need to multiply the two functions \( f(n) \) and \( h(n) \).

\[
(f \cdot h)(n) = f(n) \cdot h(n)
\]

Substitute the given functions:

\[
f(n) = -10n - 5
\]
\[
h(n) = 5n^2 - 9
\]

Now, multiply them:

\[
(f \cdot h)(n) = (-10n - 5)(5n^2 - 9)
\]

Use the distributive property (FOIL method for polynomials):

\[
= (-10n)(5n^2) + (-10n)(-9) + (-5)(5n^2) + (-5)(-9)
\]

Calculate each term:

\[
= -50n^3 + 90n - 25n^2 + 45
\]

Combine like terms (there are no like terms here):

\[
(f \cdot h)(n) = -50n^3 - 25n^2 + 90n + 45
\]

Final Answer:
\[
\boxed{-50n^3 - 25n^2 + 90n + 45}
\]

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#### 2) Given:
- \( p(z) = z^3 - 7z \)
- \( f(z) = 6z + 5 \)
- Find \( \left( \frac{f}{p} \right)(z) \)

Solution:
The operation \( \left( \frac{f}{p} \right)(z) \) means we need to divide the function \( f(z) \) by the function \( p(z) \).

\[
\left( \frac{f}{p} \right)(z) = \frac{f(z)}{p(z)}
\]

Substitute the given functions:

\[
f(z) = 6z + 5
\]
\[
p(z) = z^3 - 7z
\]

So,

\[
\left( \frac{f}{p} \right)(z) = \frac{6z + 5}{z^3 - 7z}
\]

Factor the denominator \( p(z) \):

\[
z^3 - 7z = z(z^2 - 7)
\]

Thus,

\[
\left( \frac{f}{p} \right)(z) = \frac{6z + 5}{z(z^2 - 7)}
\]

Final Answer:
\[
\boxed{\frac{6z + 5}{z(z^2 - 7)}}
\]

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#### 3) Given:
- \( h(s) = 10s - 3 \)
- \( f(s) = 9s + 4 \)
- Find \( (h \circ f)(s) \)

Solution:
The operation \( (h \circ f)(s) \) means we need to compose the functions \( h \) and \( f \). This means substituting \( f(s) \) into \( h(s) \).

\[
(h \circ f)(s) = h(f(s))
\]

Substitute \( f(s) = 9s + 4 \) into \( h(s) = 10s - 3 \):

\[
h(f(s)) = h(9s + 4)
\]

Now, replace \( s \) in \( h(s) \) with \( 9s + 4 \):

\[
h(9s + 4) = 10(9s + 4) - 3
\]

Distribute and simplify:

\[
= 10 \cdot 9s + 10 \cdot 4 - 3
\]
\[
= 90s + 40 - 3
\]
\[
= 90s + 37
\]

Final Answer:
\[
\boxed{90s + 37}
\]

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#### 4) Given:
- \( h(y) = 4y + 7 \)
- \( g(y) = y^2 + 3 \)
- Find \( (g \circ h)(y - 10) \)

Solution:
The operation \( (g \circ h)(y - 10) \) means we first compose \( g \) and \( h \), then evaluate the result at \( y - 10 \).

First, find \( (g \circ h)(y) \):

\[
(g \circ h)(y) = g(h(y))
\]

Substitute \( h(y) = 4y + 7 \) into \( g(y) = y^2 + 3 \):

\[
g(h(y)) = g(4y + 7)
\]

Now, replace \( y \) in \( g(y) \) with \( 4y + 7 \):

\[
g(4y + 7) = (4y + 7)^2 + 3
\]

Expand \( (4y + 7)^2 \):

\[
(4y + 7)^2 = (4y)^2 + 2 \cdot 4y \cdot 7 + 7^2
\]
\[
= 16y^2 + 56y + 49
\]

So,

\[
g(4y + 7) = 16y^2 + 56y + 49 + 3
\]
\[
= 16y^2 + 56y + 52
\]

Thus,

\[
(g \circ h)(y) = 16y^2 + 56y + 52
\]

Now, evaluate \( (g \circ h)(y) \) at \( y - 10 \):

\[
(g \circ h)(y - 10) = 16(y - 10)^2 + 56(y - 10) + 52
\]

Expand \( (y - 10)^2 \):

\[
(y - 10)^2 = y^2 - 20y + 100
\]

So,

\[
16(y - 10)^2 = 16(y^2 - 20y + 100) = 16y^2 - 320y + 1600
\]

Next, expand \( 56(y - 10) \):

\[
56(y - 10) = 56y - 560
\]

Combine all terms:

\[
(g \circ h)(y - 10) = 16y^2 - 320y + 1600 + 56y - 560 + 52
\]
\[
= 16y^2 - 264y + 1092
\]

Final Answer:
\[
\boxed{16y^2 - 264y + 1092}
\]

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Continue solving the remaining problems similarly. If you need further solutions, let me know!