Introduction to Exponential Functions: Videos & Practice Problems

Exponential functions, such as 2x, have a constant positive base and a variable exponent. To evaluate these functions, substitute values for the variable, applying exponent rules. For example, 24 equals 16, while 2-3 results in 123 or 1/8. Understanding these principles is crucial for evaluating and graphing exponential functions effectively.

Exponential Functions

Exponential functions are a fundamental concept in mathematics, distinct from polynomial functions. A basic example of an exponential function is f(x) = 2^x, where the base (2) is a constant and the exponent (x) is a variable. Understanding the characteristics of the base and exponent is crucial for identifying exponential functions. The base must be a positive constant that is not equal to 1, while the exponent must contain a variable.

For instance, in the function f(x) = (2/3)^x, the base is 2/3, which meets the criteria of being constant, positive, and not equal to 1. Therefore, this is an exponential function. Conversely, in the function f(y) = 1^y, the base is 1, which disqualifies it from being an exponential function since the base cannot equal 1. Lastly, in f(x) = 10^(x+1), the base is 10, and the exponent x + 1 contains a variable, confirming it as an exponential function.

Evaluating exponential functions involves substituting values for the variable in the exponent. For example, to evaluate f(x) = 2^x at x = 4, you calculate f(4) = 2^4 = 16. When dealing with negative exponents, such as f(-3) = 2^{-3}, it translates to 1/(2^3) = 1/8. For non-integer values, like x = 3.14, using a calculator is advisable. You would input 2, use the caret key for exponentiation, and then enter 3.14 to find that f(3.14) ≈ 8.815.

For larger exponents, such as x = 12, you can similarly use a calculator to compute f(12) = 2^{12} = 4096. This approach simplifies the evaluation process, especially when dealing with large numbers. Understanding these principles of exponential functions is essential for further studies in mathematics and its applications.

Determine if the function is an exponential function.  
If so, identify the power & base, then evaluate for $x=4$.
$f\left(x\right)=\left(-2\right)^{x}$

• A. Exponential function, $f\left(4\right)=16$
• B. Exponential function, $f\left(4\right)=-16$
• C. Not an exponential function

Answer: (C)

Determine if the function is an exponential function.  
If so, identify the power & base, then evaluate for $x=4$ .
$f\left(x\right)=3\left(1.5\right)^{x}$

• A. Exponential function, $f\left(4\right)=410.06$
• B. Exponential function, $f\left(4\right)=15.19$
• C. Not an exponential function

Answer: (B)

Determine if the function is an exponential function.  
If so, identify the power & base, then evaluate for $x=4$ .
$f\left(x\right)=\left(\frac12\right)^{x}$

• A. Exponential function, $f\left(4\right)=\frac{1}{16}$
• B. Exponential function, $f\left(4\right)=-16$
• C. Not an exponential function

Answer: (A)

Here’s what students ask on this topic:

What is an exponential function and how is it different from a polynomial function?

An exponential function is a mathematical function of the form f(x) = a^x, where 'a' is a constant base and 'x' is a variable exponent. This is different from a polynomial function, which has the form f(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀, where the exponents are constants and the coefficients are variables. In exponential functions, the variable is in the exponent, whereas in polynomial functions, the variable is the base raised to a constant power.

How do you evaluate an exponential function for a given value of x?

To evaluate an exponential function for a given value of x, substitute the value of x into the function and apply the rules of exponents. For example, if the function is f(x) = 2^x and you need to evaluate it for x = 4, you would calculate 2⁴. Using exponent rules, 2⁴ = 2 × 2 × 2 × 2 = 16. For non-integer exponents or large values, use a calculator to find the result.

What are the criteria for a function to be considered an exponential function?

For a function to be considered an exponential function, it must meet the following criteria: the base must be a constant, positive number, and it cannot be equal to 1. The exponent must contain a variable. For example, f(x) = 2^x is an exponential function because the base (2) is constant, positive, and not equal to 1, and the exponent (x) is a variable.

How do you graph an exponential function?

To graph an exponential function, follow these steps: 1) Identify the base and the exponent. 2) Create a table of values by choosing several values for x and calculating the corresponding y values. 3) Plot these (x, y) points on a coordinate plane. 4) Draw a smooth curve through the points. Exponential functions typically show rapid growth or decay, depending on whether the base is greater than 1 (growth) or between 0 and 1 (decay).

What happens when the exponent in an exponential function is negative?

When the exponent in an exponential function is negative, the function represents a reciprocal. For example, if f(x) = 2^-3, this is equivalent to 1/(2³). Using exponent rules, 2³ = 2 × 2 × 2 = 8, so 2^-3 = 1/8. Negative exponents indicate that the base is in the denominator of a fraction.