Let’s start by noting that the functions f and g given by

f(x) = 2^{x} and g(x) = x^{2}

are not the same function. Whether a variable appears as an exponent with a constant base or as a base with a constant exponent makes a big difference. The function *g* is a quadratic function, which we have already discussed. The function *f* is an* exponential function*. The graphs of* f* and *g* are shown in Figure 1. As expected, they are very different.

We know how to define the values of 2^{x }for many types of inputs. For positive integers, it’s simply repeated multiplication:

2^{2 }= 2 ∙ 2 = 4; 2^{3 }= 2 ∙ 2 ∙ 2 = 8; 2^{4 }= 2 ∙ 2 ∙ 2 ∙ 2 = 16

For negative integers, we use properties of negative exponents:

For rational numbers, a calculator comes in handy:

The only catch is that we don’t know how to define 2^{x }for *all* real numbers. For example, what does

2^{√2}

mean? Your calculator can give you a decimal approximation, but where does it come from? That question is not easy to answer at this point. In fact, a precise definition of 2^{√2} must wait for more advanced courses. For now, we will simply state that for any positive real number b, the expression b^{x} is defined for all real values of x, and the output is a real number as well. This enables us to draw the continuous graph for* f(x)* = 2^{x} in Figure 1. In Problems 79 and 80 in Exercises 5-1, we will explore a method for defining* b ^{x} f*or irrational x values like 2

^{√2}.

**DEFINITION 1 Exponential Function**

The equation

* f(x)* =* b ^{x } b *> 0, b ≠ 1

defines an **exponential function** for each different constant b, called the **base**.

The independent variable x can assume any real value. The domain of f is the set of all real numbers, and it can be shown that the range of* f* is the set of all positive real numbers. We require the base b to be positive to avoid imaginary numbers such as (-2)^{1/2} Problems 53 and 54 in Exercises 5-1 explore why b = 0 and b = 1 are excluded.