Identify the base and exponent of the following:

a) $f(x) = e^{2x+3}$

b) $f(x) = (1/2)^x$

Solution:

a) Base $= e$, Exponent $= 2x + 3$

b) Base $= 1/2$, Exponent $= x$

Solve for $x$: $3^{x^2-5x+2} = 9^{x+1}$

Solution:

$3^{x^2-5x+2} = (3^2)^{x+1}$

$3^{x^2-5x+2} = 3^{2x+2}$

Since the bases are equal, equate the exponents:

$x^2 - 5x + 2 = 2x + 2$

$x^2 - 7x = 0$

$x(x - 7) = 0$

$x = 0$ or $x = 7$

When drawing graphs of $f(x) = a^x$, if $a > 1$, the function increases as $x$ increases (exponential growth). If $0 < a < 1$, the function decreases as $x$ increases (exponential decay).

The graphs of $f(x) = a^x$ ($a > 0$) contain the point $(0, 1)$ because $a^0 = 1$ (y-intercept). There is no x-intercept because the graph does not cross the x-axis. The x-axis ($y = 0$) is the horizontal asymptote.

Draw on the same axes the graphs of:

a) $f(x) = 3^x$

b) $g(x) = (1/3)^x$

Solution:

Table of values:

The domain of $a^x$ is $\{x : x \in \mathbb{R}\}$ (all real numbers), and the range is $\{y : y > 0\}$ (all positive real numbers).

State the domain and range of:

a) $f(x) = 3^x$

b) $g(x) = (1/3)^x$

Solution:

a) Domain: $\{x : x \in \mathbb{R}\}$, Range: $\{f(x) : f(x) > 0\}$

b) Domain: $\{x : x \in \mathbb{R}\}$, Range: $\{g(x) : g(x) > 0\}$