Question22. If $f(x)=\frac{1}{1+x}$ and $g(x)=\frac{1}{2+x}$ , determine $f(g(x))$ .

Studdy Solution

STEP 1

1. We are given two functions $f(x) = \frac{1}{1+x}$ and $g(x) = \frac{1}{2+x}$ .
2. We need to find the composition of these functions, specifically $f(g(x))$ .

STEP 2

1. Understand the composition of functions.
2. Substitute $g(x)$ into $f(x)$ .
3. Simplify the expression for $f(g(x))$ .

STEP 3

To find $f(g(x))$ , we need to understand that this means we will substitute the entire function $g(x)$ into every instance of $x$ in the function $f(x)$ .

STEP 4

Substitute $g(x) = \frac{1}{2+x}$ into $f(x) = \frac{1}{1+x}$ . This means replacing $x$ in $f(x)$ with $g(x)$ :
$f(g(x)) = f\left(\frac{1}{2+x}\right) = \frac{1}{1 + \frac{1}{2+x}}$

STEP 5

Simplify the expression $\frac{1}{1 + \frac{1}{2+x}}$ . To do this, we need to simplify the denominator:
1. Start with the expression inside the denominator: $1 + \frac{1}{2+x}$ .
2. Find a common denominator for the terms in the denominator:
$1 + \frac{1}{2+x} = \frac{(2+x)}{(2+x)} + \frac{1}{2+x} = \frac{(2+x) + 1}{2+x} = \frac{3+x}{2+x}$
3. Substitute back into the main expression:
$f(g(x)) = \frac{1}{\frac{3+x}{2+x}}$
4. Simplify by taking the reciprocal of the fraction in the denominator:
$f(g(x)) = \frac{2+x}{3+x}$
The expression for $f(g(x))$ is $\boxed{\frac{2+x}{3+x}}$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms