Daniaal Sanchez

2021-08-17

Let $y=f\left(x\right)$ be a function D=(-12,13) and range Find the domain D and range R for the following functions and enter your answers using interval notation. Keep in mind order of operations. Be sure your intervals are in the correct order, and enter exact answers only (no approximations).
a) $y=f\left(\frac{1}{2}x\right)$
b) $y=5f\left(2x\right)-2$

Neelam Wainwright

Step 1
As we know:
Domain represent the set of inputs for a function.
Raange represent the set of outputs for a function.
When a function is transformed, its domain and/or range will change
If only inputs are transformed, then only the domain will change.If only the outputs are transformed, then only the range will change.If both the inputs and outputs are transformed, then both the domain and range will change.
For Example:
Let $f\left(x\right)$ is function with Domain and Range
1) If $f\left(x\right)\to f\left(mx\right)$ than Domain and Range will be unchanged that is Range
2) If $f\left(x\right)\to f\left(\frac{x}{n}\right)$ the Domain and Range will be unchanged that is Range
3) If $f\left(x\right)\to mf\left(x\right)$ than Domain and Range
4) If $f\left(x\right)\to f\left(x\right)±m$ than Domain and Range
5) If and Range $=\left(c,d\right)\to \left(mc,md\right)$
6) If $f\left(x\right)\to mf\left(nx\right)±p$ than Domain $=\left(a,b\right)\to \left(na,nb\right)$ and Range $=\left(c,d\right)\to \left(mc,md\right)\to \left(mc±p,md±p\right)$
Step 2
a) $y=f\left(\frac{1}{2}x\right)$
Here,Only Inputs are transformed, then only domain will change.
$⇒$ (inputs, outputs)
Since the outputs of this function are not being changed with the transformation $y=f\left(\frac{1}{2}x\right)$, that means the range is also not being changed.
So, The Range will remain
Domain: $\left(-12,13\right)\to \left(-\frac{12}{2},\frac{13}{2}\right)=\left(-6,\frac{13}{2}\right)$
So,
Domain: $\left(-6,\frac{13}{2}\right)$
Range: $\left(-\mathrm{\infty },\mathrm{\infty }\right)$
Step 3
b) Given $y=5f\left(2x\right)-2$
Here
Both the inputs and outputs are transformed,then both domain and range will change
$⇒$ (inputs, outputs)
Domain:

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