Modeling Complex Scenarios: Background You’ll Need 1

  • Recognize and evaluate exponential functions

Welcome to the fascinating world of exponential and logarithmic functions! These functions are not just abstract mathematical concepts; they are vital tools that help us understand and navigate the complexities of the world around us.

Exponential Functions

An exponential function is expressed in the form [latex]f(x)=a⋅b^x[/latex], where [latex]a[/latex] is a constant, [latex]b[/latex] is the base of the exponential, and [latex]x[/latex] is the exponent. In these functions, the variable [latex]x[/latex] is in the exponent, unlike linear functions where it is the base.

exponential function

For any real number [latex]x[/latex], an exponential function is a function with the form:

[latex]f(x)=ab^x[/latex]

where,

  • [latex]a[/latex] is a non-zero real number called the initial value and
  • [latex]b[/latex] is any positive real number such that [latex]b≠1[/latex].

Which of the following equations are not exponential functions?

  • [latex]f(x)=4^{3(x-2)}[/latex]
  • [latex]g(x)=x^3[/latex]
  • [latex]h(x)=(\frac{1}{3})^x[/latex]
  • [latex]j(x)=(-2)^x[/latex]

Evaluating Exponential Functions

Why do we limit the base [latex]b[/latex] to positive values? To ensure that the outputs will be real numbers. Observe what happens if the base is not positive:

  • Let [latex]b=−9[/latex] and [latex]x=\frac{1}{2}[/latex]. Then [latex]f(x)=f(\frac{1}{2})=(−9)^\frac{1}{2}=\sqrt{−9}[/latex], which is not a real number.

Why do we limit the base to positive values other than [latex]1[/latex]? Because base [latex]1[/latex] results in the constant function. Observe what happens if the base is [latex]1[/latex]:

  • Let [latex]b=1[/latex]. Then [latex]f(x)=1^x=1[/latex] for any value of [latex]x[/latex].

To evaluate an exponential function with the form [latex]f(x)=b^x[/latex], we simply substitute [latex]x[/latex] with the given value, and calculate the resulting power. For example: Let [latex]f(x)=2^x[/latex]. What is [latex]f(3)[/latex]?

[latex]\begin{array}{rcl} f(x) & = & 2^x \\ f(3) & = & 2^3 & \quad \text{Substitute } x = 3. \\ & = & 8 & \quad \text{Evaluate the power.} \end{array}[/latex]

 

To evaluate an exponential function with a form other than the basic form, it is important to follow the order of operations. For example: Let [latex]f(x)=30(2)^x[/latex]. What is [latex]f(3)[/latex]?

[latex]\begin{array}{rcll} f(x) & = & 30(2)^x & \\ f(3) & = & 30(2)^3 & \quad \text{Substitute } x = 3. \\ & = & 30(8) & \quad \text{Simplify the power first.} \\ & = & 240 & \quad \text{Multiply.} \end{array}[/latex]

 

Note that if the order of operations were not followed, the result would be incorrect:

[latex]f(3)=30(2)^3≠60^3=216,000[/latex]

How To: Evaluating Exponential Functions

  1. Given an exponential function, identify [latex]a[/latex], [latex]b[/latex], and the value of [latex]x[/latex] you’re being asked to substitute into the function.
  2. Replace the variable [latex]x[/latex] in the function with the given number.
  3. Compute the value of [latex]b^x[/latex]. This means raising the base [latex]b[/latex] to the power of [latex]x[/latex].
  4. If there is a coefficient [latex]a[/latex] in front of the base, multiply the result of [latex]b^x[/latex] by [latex]a[/latex]. If [latex]a[/latex] is [latex]1[/latex], this step does not change the value.
  5. Simplify the expression if necessary. This could involve performing any additional multiplication or addition/subtraction if the function has more terms.
  6. Verify your result by ensuring all mathematical operations have been performed correctly and that the function has been simplified fully.

Let [latex]f(x)=5(3)^x+1[/latex]. Evaluate [latex]f(2)[/latex] without using a calculator.