8.1 Exponential Functions and Their Properties

Exponential functions

Exponential functions describe quantities that multiply by a constant factor over equal intervals. They show up everywhere: compound interest, population growth, radioactive decay, and more. The key idea is that the rate of change itself changes, which is what makes exponential behavior so different from linear behavior.

Definition and components

An exponential function has the form:

$f(x) = a \cdot b^x$

where $a \neq 0$, $b > 0$, and $b \neq 1$.

Each piece plays a specific role:

• $a$ is the initial value. It's the output when $x = 0$, since $b^0 = 1$, giving you $f(0) = a$. This is also the y-intercept.
• $b$ is the base. It determines the constant multiplier applied each time $x$ increases by 1.
• $x$ is the exponent, typically representing time or some other independent variable.

The domain is all real numbers (you can plug in any $x$). The range depends on the sign of $a$: if $a > 0$, the range is $(0, \infty)$; if $a < 0$, the range is $(-\infty, 0)$. For this course, you'll mostly work with $a > 0$.

Behavior and properties

The base $b$ controls whether the function grows or shrinks:

• Exponential growth: When $b > 1$, the function increases as $x$ increases. Larger bases mean faster growth. For example, $f(x) = 3^x$ grows faster than $f(x) = 2^x$.
• Exponential decay: When $0 < b < 1$, the function decreases as $x$ increases. The output shrinks toward zero but never reaches it. For example, $f(x) = (0.5)^x$ halves each time $x$ increases by 1.

The y-intercept is always $(0, a)$.

Every exponential function has a horizontal asymptote at $y = 0$ (the x-axis). The graph approaches this line but never crosses it, because $b^x$ is always positive.

Evaluating exponential expressions

Evaluating exponential functions

To evaluate an exponential function at a specific input, substitute the value for $x$ and simplify using order of operations (PEMDAS). Compute the exponent first, then multiply by $a$.

• Example: Given $f(x) = 2 \cdot 3^x$, find $f(2)$
  • $f(2) = 2 \cdot 3^2 = 2 \cdot 9 = 18$
• Example: Given $g(x) = 5 \cdot (0.4)^x$, find $g(3)$
  • $g(3) = 5 \cdot (0.4)^3 = 5 \cdot 0.064 = 0.32$

Simplifying expressions with exponents

These exponent rules come up constantly when working with exponential functions. Know them cold:

• Product rule: $a^m \cdot a^n = a^{m+n}$
• Quotient rule: $\frac{a^m}{a^n} = a^{m-n}$ $(a \neq 0)$
• Power rule: $(a^m)^n = a^{mn}$
• Zero exponent rule: $a^0 = 1$ $(a \neq 0)$
• Negative exponent rule: $a^{-n} = \frac{1}{a^n}$ $(a \neq 0)$

Rational exponents connect exponents to radicals. The key relationship is:

$a^{\frac{1}{n}} = \sqrt[n]{a}$

More generally, $a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$.

• Example: $8^{\frac{1}{3}} = \sqrt[3]{8} = 2$
• Example: $27^{\frac{2}{3}} = (\sqrt[3]{27})^2 = 3^2 = 9$

You can also evaluate expressions with multiple variables by substituting and then applying exponent rules:

• Example: Given $x = 2$ and $y = 3$, evaluate $2x^2y^3$
  • $2(2)^2(3)^3 = 2 \cdot 4 \cdot 27 = 216$

A common mistake here: the 2 in front is a coefficient, not a base being raised to a power. Only $x$ and $y$ have exponents.

Graphing exponential functions

Creating graphs

To graph an exponential function by hand:

1. Build a table of values. Choose several $x$-values (including negative values, zero, and positive values).
2. Substitute each $x$-value into the function and calculate $y$.
3. Plot the points on a coordinate plane.
4. Connect the points with a smooth curve. Don't use straight line segments.
5. Draw a dashed line along the horizontal asymptote ($y = 0$) to show the boundary the curve approaches.

For $f(x) = 2^x$, a useful table looks like this:

Notice how the outputs double each time $x$ increases by 1. That's the constant multiplier (base = 2) at work.

Analyzing graphs

When you look at the graph of an exponential function, identify these features:

• Y-intercept: The point $(0, a)$.
• Horizontal asymptote: The line $y = 0$. The curve gets infinitely close but never touches it.
• End behavior: For growth ($b > 1$), as $x \to \infty$, $f(x) \to \infty$, and as $x \to -\infty$, $f(x) \to 0$. For decay ($0 < b < 1$), it's reversed.
• Always increasing or always decreasing: Exponential functions are one-to-one. Growth functions only go up; decay functions only go down.

Comparing $f(x) = 2^x$ and $g(x) = \left(\frac{1}{2}\right)^x$ on the same axes is a great exercise. These two functions are reflections of each other across the y-axis, since $\left(\frac{1}{2}\right)^x = 2^{-x}$. One grows to the right while the other decays, but both pass through $(0, 1)$.

Properties of exponents for problem-solving

Solving exponential equations

When both sides of an equation can be written with the same base, you can solve by setting the exponents equal. This is the most direct method:

1. Rewrite each side as a power of the same base.
2. Set the exponents equal (since if $b^m = b^n$, then $m = n$, provided $b > 0$ and $b \neq 1$).
3. Solve for the variable.

• Example: Solve $2^x = 32$

  1. Rewrite 32 as a power of 2: $32 = 2^5$
  2. So $2^x = 2^5$
  3. Therefore $x = 5$

• Example: Solve $9^{x-1} = 27$

  1. Rewrite with base 3: $(3^2)^{x-1} = 3^3$
  2. Apply the power rule: $3^{2(x-1)} = 3^3$
  3. Set exponents equal: $2(x-1) = 3$, so $2x - 2 = 3$, giving $x = \frac{5}{2}$

When you can't rewrite with a common base, logarithms become necessary. You'll work with that technique more in upcoming sections.

Modeling with exponential functions

Exponential functions model real-world situations where a quantity multiplies by a fixed factor over equal time intervals.

The standard growth/decay model is:

$A = A_0 \cdot b^t$

where $A_0$ is the starting amount, $b$ is the growth factor per time period, and $t$ is the number of time periods.

The continuous growth/decay model uses the constant $e \approx 2.71828$:

$A = A_0 e^{kt}$

where $k > 0$ represents continuous growth and $k < 0$ represents continuous decay.

Example: A population of bacteria starts with 100 cells and doubles every 2 hours. How many bacteria will be present after 6 hours?

Since the population doubles, the growth factor is 2 per 2-hour period. In 6 hours there are $\frac{6}{2} = 3$ doubling periods.

$A = 100 \cdot 2^3 = 100 \cdot 8 = 800 \text{ bacteria}$

You can also solve this with the continuous model. First find $k$: since the population doubles in 2 hours, $2 = e^{2k}$, so $k = \frac{\ln 2}{2} \approx 0.3466$. Then:

$A = 100 \cdot e^{0.3466 \cdot 6} \approx 100 \cdot 8 = 800 \text{ bacteria}$

Both approaches give the same answer. The standard model is simpler when you know the doubling time directly; the continuous model is more flexible for problems that give you a rate.

Common real-world applications include:

• Compound interest: $A = P\left(1 + \frac{r}{n}\right)^{nt}$, where $P$ is principal, $r$ is annual rate, $n$ is compounding frequency, and $t$ is years
• Radioactive decay: Half-life problems where $b = \frac{1}{2}$ per half-life period
• Population growth: Often modeled with a percentage growth rate, so $b = 1 + r$ where $r$ is the rate as a decimal