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Math  /  Algebra

QuestionUse the functions below to find the equation for the indicated operations. f(x)=x2−2x−3g(x)=x−32x+3h(x)=3x−4f(x)=x^{2}-2 x-3 \quad g(x)=\frac{x-3}{2 x+3} \quad h(x)=3 x-4
6. g(h(x))g(h(x))

Studdy Solution

STEP 1

What is this asking? We need to find a new function, let's call it *j(x)*, by plugging *h(x)* into *g(x)*! Watch out! Remember that plugging *h(x)* into *g(x)* means replacing every *x* in *g(x)* with the entire expression for *h(x)*.
Don't forget your parentheses!

STEP 2

1. Substitute *h(x)* into *g(x)*
2. Simplify the new function

STEP 3

We have g(x)=x−32x+3g(x) = \frac{x-3}{2x+3} and h(x)=3x−4h(x) = 3x-4.
We're going to create a brand new function!

STEP 4

Remember, *g(h(x))* means every *x* in *g(x)* gets replaced by *h(x)*.
So, g(h(x))g(h(x)) becomes: g(h(x))=(3x−4)−32(3x−4)+3g(h(x)) = \frac{(3x-4) - 3}{2(3x-4) + 3} Look at how the entire expression for *h(x)*, which is 3x−43x - 4, gets plugged in wherever there's an *x* in *g(x)*.
Those parentheses are **super important** to make sure we do the math correctly!

STEP 5

In the numerator, we have (3x−4)−3(3x - 4) - 3.
Combining the **constant terms**, we get 3x−73x - 7.

STEP 6

The denominator is a bit trickier: 2(3x−4)+32(3x-4) + 3. **Distribute** the 2 to get 6x−8+36x - 8 + 3.
Now, combine the **constant terms** to get 6x−56x - 5.

STEP 7

Our simplified function is now: g(h(x))=3x−76x−5g(h(x)) = \frac{3x-7}{6x-5} Look at that, nice and neat!

STEP 8

Our final answer, the function *g(h(x))*, is: g(h(x))=3x−76x−5g(h(x)) = \frac{3x-7}{6x-5}

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