Exponential Functions: Videos & Practice Problems

Exponential functions have a constant, positive base (not equal to 1) raised to a variable power, such as $2^x$. Evaluating these functions involves substituting values for the exponent and applying exponent rules, including handling negative exponents as reciprocals. Calculators assist with non-integer or large exponents. Understanding the base and power is crucial for graphing and analyzing exponential growth or decay, distinguishing these functions from polynomials, which have variables as bases and fixed exponents. This foundational knowledge supports mastery of exponential expressions and their applications in algebra.

Evaluating 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 is x + 1, 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 evaluating at negative values, such as x = -3, the calculation becomes f(-3) = 2^{-3} = 1/(2^3) = 1/8. For non-integer values like x = 3.14, a calculator is often used: f(3.14) = 2^{3.14} ≈ 8.815. Similarly, for larger integers like x = 12, you would compute f(12) = 2^{12} = 4096.

In summary, recognizing the structure of exponential functions and how to evaluate them is essential for further studies in mathematics, particularly in fields involving growth and decay, such as biology, finance, and physics.

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

• A. Exponential function, $f\left(4\right)=16$
• B. Exponential function, $f\left(4\right)=-16$
• C. Not an exponential function
• D. Exponential function, $f\left(4\right)=-8$

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
• D. Exponential function, $f\left(4\right)=18$

Answer: (B)

Determine if the function is an exponential function.  
If so, identify the power & base, then evaluate for $x=4$
$f\left(x\right)={\left(\frac{1}{2}\right)}^x$

• A. Exponential function, $f\left(4\right)=\frac{1}{16}$
• B. Exponential function, $f\left(4\right)=-16$
• C. Not an exponential function
• D. Exponential function, $f\left(4\right)=2$

Answer: (A)

Use a calculator to evaluate the following exponential expression. Round to two decimal places.
$7^{1.3}$

• A. $12.14$
• B. $12.55$
• C. $13.24$
• D. $11.75$

Answer: (B)

Use a calculator to evaluate the following exponential expression. Round to two decimal places.
$3^{\surd 6}$

• A. $3.75$
• B. $27$
• C. $14.75$
• D. $729$

Answer: (C)

Use a calculator to evaluate the following exponential expression. Round to two decimal places.
${13}^{\pi }$

• A. $3159.05$
• B. $3001.89$
• C. $3204.76$
• D. $3107.42$

Answer: (A)

Here’s what students ask on this topic:

What is an exponential function and how do you identify its base and exponent?

An exponential function is a mathematical function where a constant, positive base (not equal to 1) is raised to a variable exponent. For example, in the function $2^x$, 2 is the base and $x$ is the exponent. To identify an exponential function, check that the base is a constant number greater than zero and not equal to 1, and the exponent contains a variable. The base can be a fraction as long as it meets these criteria. The exponent can be a single variable or an expression involving variables, such as ${10}^{x+1}$. Understanding these parts is essential for evaluating and graphing exponential functions.

How do you evaluate exponential functions for different values of the exponent?

To evaluate an exponential function, substitute the given value into the exponent and calculate the result. For example, for $2^x$ and $x=4$, compute $2^4=16$. If the exponent is negative, such as $2^{-3}$, rewrite it as the reciprocal: $\frac{1}{2^3}=rac\{1\}\{8\}$. For non-integer or large exponents like $2^{3.14}$ or $2^{12}$, use a calculator with the caret (^) key to compute the value accurately. This method ensures precise evaluation of exponential functions for any exponent.

Why can't the base of an exponential function be 1 or negative?

The base of an exponential function cannot be 1 because any number 1 raised to any power is always 1, which results in a constant function, not an exponential one. For example, $1^x=1$ for all $x$. This means there is no growth or decay, which defeats the purpose of studying exponential behavior. The base cannot be negative because raising a negative number to a variable exponent can produce undefined or complex values, especially when the exponent is not an integer. To keep the function well-defined and real-valued for all real exponents, the base must be positive and not equal to 1.

How do you use a calculator to evaluate exponential functions with non-integer exponents?

To evaluate exponential functions with non-integer exponents on a calculator, use the caret (^) key to indicate the exponent. For example, to calculate $2^{3.14}$, enter 2 ^ 3.14 on your calculator. It is helpful to use parentheses around the exponent if it is an expression, such as 2 ^ (x + 1), to ensure the calculator interprets it correctly. This method allows you to compute values that are difficult or impossible to calculate by hand, providing accurate results for any real exponent.

What is the difference between polynomial and exponential functions?

The key difference between polynomial and exponential functions lies in the placement of the variable. In polynomial functions, the variable is the base raised to a constant exponent, such as $x^2$ in $f(x)\; =x^2$. In exponential functions, the base is a constant and the variable is in the exponent, like $2^x$. This difference leads to distinct behaviors: polynomials grow at rates determined by their degree, while exponential functions grow or decay much faster depending on the base. Understanding this distinction is crucial for analyzing and graphing these functions.