Cameron Russell

2022-01-22

Let $F\left(x\right)=3x$ and $g\left(y\right)=\frac{1}{y}$, how do you find each of the compositions and domain and range?

Appohhl

Expert

For: $\left(f\circ g\right)\left(x\right)$
$h\left(x\right)={\left[3x\right]}_{x=g\left(x\right)=\frac{1}{x}}=3\cdot \frac{1}{x}=\frac{3}{x}$
For: $\left(g\circ f\right)\left(x\right)$
$r\left(x\right)={\left[\frac{1}{x}\right]}_{x=f\left(x\right)=3x}=\frac{1}{3x}$

Expert

Required:
Composite functions: a) $⇒\left(f\circ g\right)\left(x\right)$ and b) $⇒\left(g\circ f\right)\left(x\right)$
For: $\left(f\circ g\right)\left(x\right)$
Step 1
$h\left(x\right)=f\left(g\left(x\right)\right)=f\left(\frac{1}{x}\right)$
Step 2
Replace each occurrence of x in $f\left(x\right)$ with $g\left(x\right)=\frac{1}{x}$
$h\left(x\right)={\left[3x\right]}_{x=g\left(x\right)=\frac{1}{x}}=3\cdot \frac{1}{x}=\frac{3}{x}$
The dummy variable is not relevant so you can do this in terms of x or y or $\theta$
Step 3
function in simplest form no step 3 needed
For: $\left(g\circ f\right)\left(x\right)$
Step 1
$r\left(x\right)=g\left(f\left(x\right)\right)=g\left(3x\right)=$
Step 2
Replace each occurrence of x in $g\left(x\right)$ with $f\left(x\right)=3x$
$r\left(x\right)={\left[\frac{1}{x}\right]}_{x=f\left(x\right)=3x}=\frac{1}{3x}$
Step 3
function in simplest form no step 3 needed

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