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Domain and Range of Exponential and Logarithmic Functions

Recall that the domain of a function is the set of input or  $x$ -values for which the function is defined, while the range is the set of all the output or  $y$ -values that the function takes. 

A simple exponential function like  $f\left(x\right)=2^x$  has as its domain the whole real line. But its range is only the  positive  real numbers,  $y>0:f\left(x\right)$  never takes a negative value. Furthermore, it never actually reaches $0$ , though it approaches asymptotically as  $x$ goes to $-\infty$ .

If we replace   $x$ with $-x$   to get the equation $g\left(x\right)=2^{-x}$ ,  the graph gets reflected around the  $y$ -axis, but the domain and range do not change:

If we put a negative sign in frontto get the equation $h\left(x\right)=-2^x$ , the graph gets reflected around the  $x$ -axis. We still have the whole real line as our domain, but the range is now the negative numbers, $y<0$ .

Now, consider the function $f\left(x\right)={\left(-2\right)}^x$ . When $x=\frac{1}{2}$ , $y$ must be a complex number, so things get tricky. For this lesson we will require that our bases be positive for the moment, so that we can stay in the real-valued world.

In general, the graph of the basic exponential function $y=a^x$ drops from $\infty$ to $0$  when $0<a<1$  as $x$ varies from $-\infty$  to $\infty$ and rises from $0$ to $\infty$ when $a>1$ .

The exponential function $y=a^x$ , can be shifted  $k$  units vertically and  $h$  units horizontally with the equation $y=a^{\left(x+h\right)}+k$ . Then the domain of the function remains unchanged and the range becomes $\left\{y\in ℝ|y>h\right\}$ .

The inverse of an exponential function is a logarithmic function.

A simple logarithmic function $y={\log }_{2}x$  where $x>0$ is equivalent to the function $x=2^y$ . That is, $y={\log }_{2}x$ is the inverse of the function $y=2^x$ .

The function $y={\log }_{2}x$ has the domain of set of positive real numbers and the range of set of real numbers.

Remember that since the logarithmic function is the inverse of the exponential function, the domain of logarithmic function is the range of exponential function, and vice versa.

In general, the function $y={\log }_{b}x$ where $b,x>0$ and $b\ne 1$ is a continuous and one-to-one function. Note that the logarithmic functionis not defined for negative numbers or for zero. The graph of the function approaches the $y$ -axis as $x$ tends to $\infty$ , but never touches it.

Therefore, the domain of the logarithmic function $y={\log }_{b}x$ is the set of positive real numbers and the range is the set of real numbers.

The function rises from $-\infty$ to $\infty$ as $x$ increases if $b>1$ and falls from $\infty$ to $-\infty$ as $x$ increases if $0<b<1$ .

The logarithmic function, $y={\log }_{b}x$ , can be shifted  $k$  units vertically and  $h$  units horizontally with the equation $y={\log }_b\left(x+h\right)+k$ . Then the domain of the function becomes $\left\{x\in ℝ|x>-h\right\}$ . However, the range remains the same.