Math

QuestionFind the product of s(x)=x3x236s(x) = \frac{x-3}{x^{2}-36} and t(x)=x63xt(x) = \frac{x-6}{3-x}. Calculate (st)(x)(s \cdot t)(x).

Studdy Solution

STEP 1

Assumptions1. The function s(x)=x3x36s(x) = \frac{x-3}{x^{}-36} . The function t(x)=x63xt(x) = \frac{x-6}{3-x}
3. We are looking for the product of these two functions, i.e., (st)(x)(s \cdot t)(x)

STEP 2

The product of two functions s(x)s(x) and t(x)t(x), denoted as (st)(x)(s \cdot t)(x), is found by multiplying the two functions together.
(st)(x)=s(x)t(x)(s \cdot t)(x) = s(x) \cdot t(x)

STEP 3

Substitute the given functions s(x)s(x) and t(x)t(x) into the equation.
(st)(x)=(x3x236)(x63x)(s \cdot t)(x) = \left(\frac{x-3}{x^{2}-36}\right) \cdot \left(\frac{x-6}{3-x}\right)

STEP 4

We can simplify this equation by multiplying the numerators together and the denominators together.
(st)(x)=(x3)(x6)(x236)(3x)(s \cdot t)(x) = \frac{(x-3)(x-6)}{(x^{2}-36)(3-x)}

STEP 5

Notice that the denominator has a factor of (3x)(3-x) and the numerator has a factor of (x)(x-). These are negatives of each other, so they can be cancelled out.
(st)(x)=(x3)(x236)(s \cdot t)(x) = \frac{(x-3)}{(x^{2}-36)}

STEP 6

The denominator can be factored as (x6)(x+6)(x-6)(x+6).
(st)(x)=(x3)(x6)(x+6)(s \cdot t)(x) = \frac{(x-3)}{(x-6)(x+6)}

STEP 7

This is the simplified form of the product of the functions s(x)s(x) and t(x)t(x).
So, (st)(x)=(x3)(x6)(x+6)(s \cdot t)(x) = \frac{(x-3)}{(x-6)(x+6)}.

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