Question

# Graph each function. Give the domain and range. $f(x)=2^{x-1}+2$

Solution

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Given:

$f(x)=2^{x-1}+2$

The graph of $f(x)=2^{x-1}+2$ is the graph of $f(x)=2^{x}$ translated $2$ units up and $1$ unit to the right. These are from the properties of graphing vertical and horizontal translations of functions:

If $f(x)+k$ then the $f(x)$ moves $k$ units up while if $f(x)-k$ then the $f(x)$ moves $k$ units down.

If $f(x-h)$ then the $f(x)$ moves $h$ units to the right while if $f(x+h)$ then the $f(x)$ moves $h$ units to the left.

Thus, we have the graph of $f(x)=2^{x}$ moved $2$ units up and $1$ unit to the right into $f(x)=2^{x-1}+2$:

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