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1.3 Inverse, Exponential and Logarithmic Functions

\( y = f(x) \)

what is the output \( y \) when we know the input \( x \)

the inverse is

\( x = f^{-1}(y) \)

what is the input \( x \) that got me a specific output \( y \)

for example,

\( f(x) = x^3 \)

\( f(2) = 2^3 = 8 \)

input of 2 gives output 8

\( 8 = x^3 \)

\( x = 8^{1/3} = 2 \)

what input \( x \) gave us output 8

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not all functions have inverses

a function must be one-to-one, at least on some portion of its domain to have an inverse.

one-to-one: each output \( (y) \) is paired with at most one input \( (x) \)

last example: \( f(x) = x^3 \)

\( 8 = x^3 \quad \rightarrow \quad x = 2 \text{ only} \)

only possible input to get 8 is 2

so, \( f(x) = x^3 \) is one-to-one

\( f(x) = x^2 \)

is not one-to-one because output of 25

\( 25 = x^2 \quad \rightarrow \quad x = \pm 5 \)

output of 25 paired with more than one input

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The graph of a one-to-one function must pass the horizontal line test: each horizontal line drawn can only intersect the graph at most once.

A graph of an increasing curve intersected by multiple horizontal lines, each crossing only once.

Each horizontal line intersects graph at most once.

This is the graph of a one-to-one function.

A graph of a parabola intersected by a horizontal line that crosses the curve at two points.

At least one horizontal line crosses the graph more than once.

This is NOT the graph of a one-to-one function.

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Even for a function that is not one-to-one, we can restrict its domain to make it one-to-one.

For example, \( f(x) = x^2 \) is not one-to-one on \( (-\infty, \infty) \).

Graph of a full parabola y=x^2 with a horizontal line intersecting it at two points.

But it is one-to-one if we only consider \( [0, \infty) \).

Graph of the right half of a parabola y=x^2 with horizontal lines intersecting it only once.

\( f(x) = |x - 2| \)

Graph of an absolute value function y=|x-2| showing a V-shape intersected twice by a horizontal line.

NOT one-to-one on \( (-\infty, \infty) \)

But is one-to-one on \( [2, \infty) \)

Graph of the right half of y=|x-2| starting from x=2, intersected once by a horizontal line.
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Inverse Functions and One-to-One Intervals

If \( f(x) \) is one-to-one on at least some interval, then we can find its inverse on some parts of it.

Example

\[ f(x) = \frac{1}{x-1} \]

Is it one-to-one?

Domain: \( (-\infty, 1), (1, \infty) \)

Passes the horizontal line test.

So it is one-to-one, so it has an inverse.

Graph of f(x) = 1/(x-1) showing two branches and horizontal red lines intersecting only once.

Find the Inverse (as an example, on \( (1, \infty) \))

\[ y = \frac{1}{x-1} \]

We interchange \( x \) and \( y \) (because we know output already and want the input):

\[ x = \frac{1}{y-1} \]

Then solve for \( y \).

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Multiply both sides by \( y-1 \) and divide by \( x \):

\[ y-1 = \frac{1}{x} \]
\[ y = \frac{1}{x} + 1 \]

Then rewrite \( y \) as \( f^{-1}(x) \):

\[ f^{-1}(x) = \frac{1}{x} + 1 \]

\( 0 < x < \infty \)

Output is positive (right part of graph)

The Relationship Between Graphs of \( f(x) \) and \( f^{-1}(x) \)

Coordinate graph showing f(x) in green, f inverse in black, and reflection line y=x dashed.

\( x \) and \( y \) switch places on \( f^{-1}(x) \).

Graphs of \( f(x) \) and \( f^{-1}(x) \) are reflections of each other across the line \( y = x \).

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Exponential and Logarithmic Functions are Inverses of Each Other

Exponential

\[ y = b^x \]
\[ f(x) = b^x \]

Logarithmic

\[ x = \log_b y \]
\[ f^{-1}(x) = \log_b x \]

Example

\[ f(x) = 3^x \]
\[ f(2) = 3^2 = 9 \]

Output is 9 when input is 2

\[ f^{-1}(x) = \log_3 x \]

output

What input gave us an output of 27?

\[ f^{-1}(x) = \log_3 27 = \log_3 3^3 = 3 \log_3 3 = 3 \]
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Useful Properties of Logarithm

\[ \log_b(xy) = \log_b x + \log_b y \]
\[ \log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y \]
\[ \log_b x^c = c \log_b x \]
\[ \log_b 1 = 0 \]
\[ \log_b b = 1 \]
\[ b^{\log_b x} = x \]
\[ \log_b b^x = x \]
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Logarithm Properties and Exponential Equations

For example,

\[ \log_3 \frac{\sqrt{x}}{\sqrt[5]{y}} \]
\[ = \log_3 \sqrt{x} - \log_3 \sqrt[5]{y} \quad \text{(2nd property)} \]
\[ = \log_3 x^{1/2} - \log_3 y^{1/5} \]
\[ = \frac{1}{2} \log_3 x - \frac{1}{5} \log_3 y \quad \text{(3rd property)} \]

Solving Exponential Equations

The properties can also be used to solve exponential equations.

For example,

\[ 2^{5x-3} = 18 \quad x = ? \]

Take log of any base on both sides.

Usually base \( e \) or base 10.

\( \log_e = \ln \)

\( \log_{10} = \log \)

\[ \ln 2^{5x-3} = \ln 18 \]
\[ (5x-3) \cdot (\ln 2) = \ln 18 \]
\[ 5x - 3 = \frac{\ln 18}{\ln 2} \rightarrow 5x = 3 + \frac{\ln 18}{\ln 2} \]
\[ x = \frac{3}{5} + \frac{\ln 18}{5 \ln 2} \]