Functions

Exponential Functions

Exponential functions are of the form $y = a^x$ , where $a > 0$ and $a \ne 1$ and $x$ is any real number. You might see these sorts of functions when studying population growth, economic growth, global temperature, monetary value, etc.

Change the base and move the input. Compare the selected curve with its reciprocal-base partner: growth becomes decay.

\displaystyle y=2^x

\displaystyle y=\left(\frac{1}{2}\right)^x

Base a2Input x0

\displaystyle f(0)=1

a > 1: increasing · y(0) = 1 · y(1) = 2.0

Domain: $x \in \mathbb{R}$
Range: $y > 0$
The graph $y = a^x$ always passes through $(0, 1)$ and $(1, a)$
If $a > 1$ then the graph of $y = a^x$ is increasing.
If $0 < a < 1$ then the graph of $y = a^x$ is decreasing.
$y = 0$ is always a horizontal asymptote of $y = a^x$

QUESTION

What is true about

\displaystyle y=\left(\dfrac12\right)^x

Exponent Rules

THEOREM

Exponent Rules

\begin{aligned} & a^{-x} = {\dfrac{1}{a^x}} \\[0.65em] & {\dfrac{1}{a^{-x}}} = a^x \\[0.65em] & (ab)^x = a^xb^x \\[0.65em] & ({\dfrac{a}{b}})^x = {\dfrac{a^x}{b^x}} \\[0.65em] & a^{kx} = (a^k)^x = (a^x)^k \\[0.65em] & a^ma^n = a^{m+n} \\[0.65em] & {\dfrac{a^m}{a^n}} = a^{m-n} \\[0.65em] & a^{1/n} = \sqrt[n]{a} \\[0.65em] & a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m \end{aligned}

The Base e

A very special exponential function is $y = e^x$ , where $e$ is just a constant with a non-terminating decimal like $\pi$

e = 2.718281828459045\ldots

What is so special about an exponential function with base $e$ ?

At any point on the graph, the height of the exponential function is equal to the slope of the tangent line to the graph at that point.

Move the point of tangency. The tangent slope and the curve's height are both eˣ at that point.

\displaystyle y=e^x

\displaystyle y=e^{0}(x-(0))+e^{0}

Point x0

\displaystyle f(0)=1

Tangent slope: 1.000

The excluded base $a=1$ gives the constant function $y=1$ For the exponent rules above, take $a,b>0$ ; root indices are positive integers.

◀ Previous SectionSpecial Triangles and Trigonometric Identities Next Section ▶Logarithmic Functions and Equations

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Functions

Exponential Functions

Change the base and move the input. Compare the selected curve with its reciprocal-base partner: growth becomes decay.

\displaystyle y=2^x

\displaystyle y=\left(\frac{1}{2}\right)^x

Base a2Input x0

\displaystyle f(0)=1

a > 1: increasing · y(0) = 1 · y(1) = 2.0

Domain: $x \in \mathbb{R}$
Range: $y > 0$
The graph $y = a^x$ always passes through $(0, 1)$ and $(1, a)$
If $a > 1$ then the graph of $y = a^x$ is increasing.
If $0 < a < 1$ then the graph of $y = a^x$ is decreasing.
$y = 0$ is always a horizontal asymptote of $y = a^x$

QUESTION

What is true about

\displaystyle y=\left(\dfrac12\right)^x

Exponent Rules

THEOREM

Exponent Rules

\begin{aligned} & a^{-x} = {\dfrac{1}{a^x}} \\[0.65em] & {\dfrac{1}{a^{-x}}} = a^x \\[0.65em] & (ab)^x = a^xb^x \\[0.65em] & ({\dfrac{a}{b}})^x = {\dfrac{a^x}{b^x}} \\[0.65em] & a^{kx} = (a^k)^x = (a^x)^k \\[0.65em] & a^ma^n = a^{m+n} \\[0.65em] & {\dfrac{a^m}{a^n}} = a^{m-n} \\[0.65em] & a^{1/n} = \sqrt[n]{a} \\[0.65em] & a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m \end{aligned}

The Base e

A very special exponential function is $y = e^x$ , where $e$ is just a constant with a non-terminating decimal like $\pi$

e = 2.718281828459045\ldots

What is so special about an exponential function with base $e$ ?

At any point on the graph, the height of the exponential function is equal to the slope of the tangent line to the graph at that point.

Move the point of tangency. The tangent slope and the curve's height are both eˣ at that point.

\displaystyle y=e^x

\displaystyle y=e^{0}(x-(0))+e^{0}

Point x0

\displaystyle f(0)=1

Tangent slope: 1.000

The excluded base $a=1$ gives the constant function $y=1$ For the exponent rules above, take $a,b>0$ ; root indices are positive integers.

◀ Previous SectionSpecial Triangles and Trigonometric Identities Next Section ▶Logarithmic Functions and Equations

Exponential Functions