Definition 7.2.1. Exponential Function.
An exponential function has the form
\begin{equation*}
\blert{f(x) = ab^x},~~~~ \text{ where } ~~~b \gt 0 ~~~\text{ and } ~~~b \ne 1 \text{, } ~~~a \ne 0
\end{equation*}
\(x\) | \(f(x)\) |
\(-3\) | \(\frac{1}{8}\) |
\(-2\) | \(\frac{1}{4}\) |
\(-1\) | \(\frac{1}{2}\) |
\(0\) | \(1\) |
\(1\) | \(2\) |
\(2\) | \(4\) |
\(3\) | \(8\) |
\(x\) | \(g(x)\) |
\(-3\) | \(8\) |
\(-2\) | \(4\) |
\(-1\) | \(2\) |
\(0\) | \(1\) |
\(1\) | \(\frac{1}{2}\) |
\(2\) | \(\frac{1}{4}\) |
\(3\) | \(\frac{1}{8}\) |
\(x\) | \(f(x)\) | \(g(x)\) |
\(-2\) | \(\dfrac{1}{9}\) | \(\dfrac{1}{16}\) |
\(-1\) | \(\dfrac{1}{3}\) | \(\dfrac{1}{4}\) |
\(0\) | \(1\) | \(1\) |
\(1\) | \(3\) | \(4\) |
\(2\) | \(9\) | \(16\) |
\(\hphantom{General formula and m}\) |
Power Functions |
Exponential Functions |
General formula |
\(h(x)=kx^p\) |
\(f(x)=ab^x\) |
Description |
variable base and constant exponent |
constant base and variable exponent |
Example |
\(h(x)=2x^3\) |
\(f(x)=2(3^x)\) |
\(x\) | \(h(x)\) |
\(-3\) | \(-54\) |
\(-2\) | \(-16\) |
\(-1\) | \(-2\) |
\(0\) | \(0\) |
\(1\) | \(2\) |
\(2\) | \(16\) |
\(3\) | \(54\) |
\(x\) | \(f(x)\) |
\(-3\) | \(\frac{2}{27}\) |
\(-2\) | \(\frac{1}{4}\) |
\(-1\) | \(\frac{2}{3}\) |
\(0\) | \(2\) |
\(1\) | \(6\) |
\(2\) | \(18\) |
\(3\) | \(54\) |
\(t\) | \(P(t)\) | \(\hphantom{0000}\) |
\(0\) | \(10\) | \(\hphantom{0000}\) |
\(5\) | \(20\) | \(P(\alert{5}) = 10 \cdot 2^{\alert{1}} = 20\) |
\(10\) | \(40\) | \(P(\alert{10}) = 10 \cdot 2^{\alert{2}} = 40\) |
\(15\) | \(80\) | \(P(\alert{15}) = 10 \cdot 2^{\alert{3}} = 80\) |
\(20\) | \(160\) | \(P(\alert{20}) = 10 \cdot 2^{\alert{4}} = 160\) |
\(x\) | \(-3\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) | \(3\) |
\(f(x)=3^x \) | \(\frac{1}{27} \) | \(\frac{1}{9} \) | \(\frac{1}{3} \) | \(1\) | \(3\) | \(9\) | \(27\) |
\(g(x)=\left(\frac{1}{3} \right)^x \) | \(27\) | \(9\) | \(3\) | \(1\) | \(\frac{1}{3} \) | \(\frac{1}{9} \) | \(\frac{1}{27} \) |
\(x\) | \(0\) | \(1\) | \(2\) |
\(g(x)\) | \(800\) | \(200\) | \(50\) |
\(h\) | \(a\) |
\(0\) | \(70\) |
\(1\) | \(7\) |
\(2\) | \(0.7\) |
\(3\) | \(0.07\) |
\(4\) | \(0.007\) |
\(t\) | \(Q\) |
\(0\) | \(0\) |
\(1\) | \(\frac{1}{4} \) |
\(2\) | \(1\) |
\(3\) | \(\frac{9}{4} \) |
\(4\) | \(4\) |
\(t\) | \(y\) |
\(1\) | \(100\) |
\(2\) | \(50\) |
\(3\) | \(33\frac{1}{3} \) |
\(4\) | \(25\) |
\(5\) | \(20\) |
\(x\) | \(P\) |
\(1\) | \(\frac{1}{2} \) |
\(2\) | \(1\) |
\(3\) | \(2\) |
\(4\) | \(4\) |
\(5\) | \(8\) |
\(x\) | \(f(x)=x^2\) | \(g(x)=2^x \) |
\(-2\) | ||
\(-1\) | ||
\(0\) | ||
\(1\) | ||
\(2\) | ||
\(3\) | ||
\(4\) | ||
\(5\) |
\(x\) | \(f(x)=x^2\) | \(g(x)=2^x \) |
\(-2\) | \(4\) | \(\frac{1}{4} \) |
\(-1\) | \(1\) | \(\frac{1}{2} \) |
\(0\) | \(0\) | 1 |
\(1\) | \(1\) | \(2\) |
\(2\) | \(4\) | \(4\) |
\(3\) | \(9\) | \(8\) |
\(4\) | \(16\) | \(16\) |
\(5\) | \(25\) | \(32\) |
\(x\) | \(f(x)=x^3\) | \(g(x)=3^x \) |
\(-2\) | ||
\(-1\) | ||
\(0\) | ||
\(1\) | ||
\(2\) | ||
\(3\) | ||
\(4\) | ||
\(5\) |
Size rank | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) |
Mass (grams) | \(49\) | \(76\) | \(123\) | \(163\) | \(245\) | \(414\) | \(592\) | \(802\) |