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Math  /  Algebra

Questionf(r)=r2f(d−6)=\begin{array}{l}f(r)=r^{2} \\ f(d-6)=\end{array}

Studdy Solution

STEP 1

What is this asking? Given a function f(r)=r2f(r) = r^2, what is f(d−6)f(d-6)? Watch out! Don't forget to square the *entire* input (d−6)(d-6), not just the dd and the 66 separately!

STEP 2

1. Substitute the Input
2. Expand the Expression
3. Simplify the Result

STEP 3

We're given the function f(r)=r2f(r) = r^2.
This tells us that whatever we put inside the parentheses gets **squared**!
Our input this time is (d−6)(d-6), so let's **plug it in**: f(d−6)=(d−6)2f(d-6) = (d-6)^2.
See how the whole thing (d−6)(d-6) is now being squared?
That's super important!

STEP 4

Now, (d−6)2(d-6)^2 really means (d−6)⋅(d−6)(d-6) \cdot (d-6).
Let's **expand** this using the distributive property (sometimes called FOIL).
So, we have: (d−6)⋅(d−6)=d⋅(d−6)+(−6)⋅(d−6)(d-6) \cdot (d-6) = d \cdot (d-6) + (-6) \cdot (d-6)

STEP 5

Distribute the dd in the first part: d⋅(d−6)=d⋅d+d⋅(−6)=d2−6dd \cdot (d-6) = d \cdot d + d \cdot (-6) = d^2 - 6d.

STEP 6

Distribute the −6-6 in the second part: (−6)⋅(d−6)=(−6)⋅d+(−6)⋅(−6)=−6d+36(-6) \cdot (d-6) = (-6) \cdot d + (-6) \cdot (-6) = -6d + 36.

STEP 7

**Combine** the results: (d2−6d)+(−6d+36)(d^2 - 6d) + (-6d + 36).

STEP 8

We can **combine** the like terms −6d-6d and −6d-6d: −6d+(−6d)=−12d-6d + (-6d) = -12d.

STEP 9

So, our **final expression** is d2−12d+36d^2 - 12d + 36!

STEP 10

f(d−6)=d2−12d+36f(d-6) = d^2 - 12d + 36

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