Math

QuestionFind f(g(x))f(g(x)) for f(x)=52x+1f(x)=\frac{5}{2 x+1} and g(x)=x23g(x)=x^{2}-3. Identify Olivia's mistake and correct her work.

Studdy Solution

STEP 1

Assumptions1. The function f(x)f(x) is given as f(x)=5x+1f(x)=\frac{5}{ x+1} . The function g(x)g(x) is given as g(x)=x3g(x)=x^{}-3
3. We are asked to find the composition of the functions, f(g(x))f(g(x))
4. Olivia's work is given as Line1 f(g(x))=f(x3)f(g(x))=f\left(x^{}-3\right) Line =5x3+1\quad=\frac{5}{ x^{}-3+1} Line3 =5x\quad=\frac{5}{ x^{}-}

STEP 2

We need to identify where Olivia made a mistake in her work. Let's start by examining Line1.
In Line1, Olivia correctly substituted g(x)g(x) into f(x)f(x), replacing xx in f(x)f(x) with x2x^{2}- from g(x)g(x).
f(g(x))=f(x2)f(g(x))=f\left(x^{2}-\right)This step is correct.

STEP 3

Now let's examine Line2.
In Line2, Olivia substituted x23x^{2}-3 into f(x)f(x), replacing xx in f(x)f(x) with x23x^{2}-3.
52x23+1\frac{5}{2 x^{2}-3+1}However, this is not correct. She should have replaced xx in f(x)f(x) with x23x^{2}-3 in the denominator, not 2x23+12x^{2}-3+1. This is where Olivia made her mistake.

STEP 4

Now let's show the correct way to solve this problem.
First, substitute g(x)g(x) into f(x)f(x), replacing xx in f(x)f(x) with x23x^{2}-3 from g(x)g(x).
f(g(x))=f(x23)f(g(x))=f\left(x^{2}-3\right)This is the same as Line1 in Olivia's work.

STEP 5

Next, substitute x23x^{2}-3 into f(x)f(x), replacing xx in f(x)f(x) with x23x^{2}-3.
f(x23)=52(x23)+1f\left(x^{2}-3\right)=\frac{5}{2\left(x^{2}-3\right)+1}

STEP 6

implify the expression in the denominator.
f(x23)=52x26+1f\left(x^{2}-3\right)=\frac{5}{2x^{2}-6+1}

STEP 7

Further simplify the expression in the denominator.
f(x23)=52x25f\left(x^{2}-3\right)=\frac{5}{2x^{2}-5}So, the correct answer is f(g(x))=52x25f(g(x))=\frac{5}{2x^{2}-5}.

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