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Precalculus Review Materials

Jane Butterfield, Gary MacGillivray

Section 3.5 Exponential Functions

Let \(b > 0\) be a positive real number. In previous sections we talked about the numbers \(b^r\text{,}\) where \(r\) is an integer or a rational number (a rational number is a fraction; a ratio of integers). In this section we consider numbers \(b^r\text{,}\) where \(r\) is any real number.
The precise definition of a number like \(3^{\sqrt{2}}\) is a topic in mathematical analysis, and is beyond the scope of these notes. Here’s a reasonable way to think about it. The number \(\sqrt{2} = 1.4142\ldots\text{.}\) Each of the numbers \(1, 1.4, 1.41, 1.414, 1.4142, \ldots\) is rational, so the numbers \(3^1\text{,}\) \(3^{1.4}\text{,}\) \(3^{1.41}\text{,}\) \(3^{1.414}\text{,}\) \(3^{1.4142}\text{,}\) \(\ldots\) are all defined. These numbers are approximately \(3, 4.6555, 4.70697, 4.727695, 4.7287, \ldots\text{.}\) It turns out that as the sequence of rational approximations gets “close” to \(\sqrt{2}\text{,}\) the sequence of exponentials gets “close” to a particular number, and that number is defined to be \(3^{\sqrt{2}}\text{.}\)
The properties of exponents are the same no matter whether the exponent is an integer, a rational number or a real number:
  • When exponential expressions with the same base are multiplied, simplify by adding the powers, i.e.,
    \begin{equation*} b^xb^y = b^{x+y} \end{equation*}
  • When an exponential expression is itself raised to some power, simplify by multiplying the exponents, i.e.,
    \begin{equation*} \left(b^x\right))^y = b^{xy} \end{equation*}
When \(b > 0\) we can talk about \(b^x\) for any real number \(x\text{.}\) (This is not true if \(b \lt 0\text{.}\) For example, remember that \((-2)^{\frac{1}{2}}\) is undefined.) Thus we can talk about the exponential function with base \(b\), \(f(x) = b^x\text{.}\)
Properties of \(f(x)=b^x\text{,}\) where \(b>0\text{.}\)
  • The domain of \(f(x)\) is the set of all real numbers.
  • Since \(b^0 = 1\text{,}\) the graph of \(f(x)=b^x\) always contains the point \((0, 1)\text{.}\)
  • There is no \(x\) such that \(b^x = 0\text{,}\) and in fact \(b^x > 0\) for every real number \(x\text{.}\)
  • If \(b > 1\text{,}\) then \(b^x\) gets larger and larger as \(x\) moves to the right on the \(x\)-axis, and gets close to zero as \(x\) moves to the left on the \(x\)-axis, and. (The first part is easy to see. To see the second part, think about the sequence of powers \(b^{-1}, b^{-2}, \ldots = \frac{1}{b}, \frac{1}{b^2}, \ldots\text{;}\) the denominators of the fractions get larger as the exponents do.)
  • If \(0 \lt b \lt 1\text{,}\) then \(b^x\) decreases towards 0 as \(x\) gets large, and gets larger and larger as \(x\) gets small. The reasoning is the similar to the reasoning above because \(\frac{1}{b} > 1\text{.}\)
  • The range of \(f(x) = b^x\) is the \((0, \infty)\text{.}\)
The graph of \(f(x) = 2^x\) is shown below.
The natural exponential function is is \(f(x) = e^x\text{,}\) where \(e \approx 2.718\text{.}\) The reasons why this function is important are best explain in calculus. The graph of \(f(x) = e^x\) is shown below.
When \(0 \lt b \lt 0\) the graph of \(f(x) = b(x)\) has a different shape. As noted above, it decreases towards 0 as \(x\) gets large, and gets larger and larger as \(x\) gets small. The graph of \(f(x) = \left(\frac{1}{2}\right)^x\) is shown below.
The principles of shifting or stretching graphs apply to the graph of any function, so in particular they apply to the graph of exponential functions.
Sketch the graph of \(f(x) = -e^{\frac{1}{3}x} + 2\text{.}\)
Solution.
The graph is similar to the graph of \(y = e^x\text{,}\) except that it is inverted (the height is scaled by -1), shifted 2 units up, stretched by a factor of 3. After plotting a few points and sketching a curve of the correct shape through them, one arrives at the graph below.
Find the positive number \(x\) such that \(x^{\sqrt{2}} = 3^5\text{.}\)
Solution.
We have \(x^{\sqrt{2}} = 3^5\text{,}\) so \(\left(x^{\sqrt{2}}\right)^{\sqrt{2}} = \left(3^5\right)^{\sqrt{2}}\text{.}\) From the properties of exponents, this is the same as \(x^2 = 3^{5 \sqrt{2}}\text{.}\) Since \(x\) is positive,
\begin{equation*} \begin{aligned} x &= \sqrt{3^{5 \sqrt{2}}} \\ &= \left(3^{5 \sqrt{2}}\right)^{\frac{1}{2}} \\ &= 3^{\frac{5}{2}\sqrt{2}} \end{aligned} \end{equation*}
The number \(3^{\frac{5}{2}\sqrt{2}} \approx 48.627\text{.}\)

Exercises Practice Problems

1.

Using \(f(x) = 4^x\text{,}\) find:
  1. \(\displaystyle f(-2)\)
  2. \(f(x)\) when \(x = 1\)
  3. \(\displaystyle f(3/2)\)
Answer.
  1. \(\displaystyle \frac{1}{16}\)
  2. \(\displaystyle 4\)
  3. \(\displaystyle 8\)

2.

Using \(f(x) = \left(1/3\right)^x\text{,}\) find:
  1. \(\displaystyle f(0)\)
  2. \(\displaystyle f(3)\)
  3. \(\displaystyle f(-3)\)
Answer.
  1. \(\displaystyle 1\)
  2. \(\displaystyle \frac{1}{27}\)
  3. \(\displaystyle 27\)

3.

Sketch the following functions on the same graph
  1. \(\displaystyle f(x)=3^x\)
  2. \(\displaystyle g(x)=(\frac{1}{3})^x\)
Answer.
The graphs are shown below.

4.

Find the positive number \(x\) such that \(x^{\sqrt{3}} = 8^2\text{.}\)
Answer.
\(x=8^{\frac{2}{3}\sqrt{3}} = 4^{\sqrt{3}}\)

5.

Sketch the graph of \(h(x)=1-3e^{x+2}\text{.}\)
Answer.
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