Exponential

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exponential and logarithmic

Exponential and logarithmic functions are examples of inverse functions.

First, let’s notice that both types of functions are one-to-one and their graphs pass the horizontal line test.

Graph of $y = E(t) = 0.5 \, e^{-t-3} - 4$ .

Graph of $z = L(x) = ln(x+5) - 1$ .

Inverse

Let $E(t) = 0.5 \, e^{-t-3} - 4$ .

Graph of $y = E(t) = 0.5 \, e^{-t-3} - 4$ .

We have a function called $E(t)$ . Its pairs look like $(a, E(a))$ . The inverse of this function will have pairs of the form $(E(a), a)$ . The domain and range just switch roles.

In the exponential formula and graph, $E$ is representing the range and $t$ the domain. They just switch roles for the inverse. Same equation. Just a different interpretation.

$E = 0.5 \, e^{-t(E)-3} - 4$

Now, $t$ is the function name and $E$ is the variable. We can just solve this for $t(E)$ .

$\begin{align*} E & = 0.5 \, e^{-t(E)-3} - 4 \\ \answer{E + 4} & = 0.5 \, e^{-t(E)-3} \\ 2(E + 4) & = e^{-t(E)-3} \\ ln(2(E+4)) & = \answer{-t(E)-3} \\ -(ln(2(E+4))) & = t(E)+3 \\ \answer{-(ln(2(E+4))) - 3} & = t(E) \end{align*}$

Graphs of $y = E(t) = 0.5 \, e^{-t-3} - 4$ and $z = L(x) = -ln(2(x+4)) - 3$ .

Since, these are inverses of each other, the pairs of each function are just reversed and the graphs are mirror images of each other over the graph of the identity function: $\{ (r,r) \, | \, r \in \mathbb {R} \}$ .

The domain of $E(t)$ is $\mathbb {R}$ . The range of $L(x)$ is $\mathbb {R}$ .
The range of $E(t)$ is $(-4, \infty )$ . The domain of $L(x)$ is $(-4, \infty )$ .

The graph of $E(t)$ has a horizontal asymptote at $-4$ .
The graph of $L(x)$ has a vertical asymptote at $-4$ .

$L(x) = ln(x+5) - 1$ is a one-to-one function, as its graph illustrates.

Therefore, it must have an inverse function, $x(L)$ .

We can just solve for $x$ .

$\begin{align*} L & = ln(x+5) - 1 \\ \answer{L + 1} & = ln(x+5) \\ \answer{e^{L+1}} & = x+5 \\ e^{L+1} - 5 & = x \\ e^{L+1} - 5 & = x(L) \end{align*}$

If the graph of $f$ has a horizontal asymptote, then the graph of $f^{-1}$ also has a horizontal asymptote.

True False

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More Examples of Function Algebra