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. Bases can be fractions, and powers may include expressions like ${10}^{x+1}$. Calculators assist with non-integer or large exponents, ensuring accurate computation of exponential expressions essential for understanding growth and decay phenomena.

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

An exponential function is defined as a function where a constant, positive base (not equal to 1) is raised to a variable exponent. For example, $2^x$ is an exponential function because the base 2 is constant and positive, and the exponent is the variable $x$. In contrast, a polynomial function has variables as bases with fixed exponents, such as $x^2$. The key difference is that in exponential functions, the variable is in the exponent, causing rapid growth or decay, while in polynomials, the variable is the base, and the exponent is a constant. This distinction affects how the functions behave and are graphed.

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

To evaluate an exponential function like $2^x$ for a specific value of $x$, substitute the value into the exponent. For example, if $x=\; 4$, then $f(4)\; =\; 2^4$. Calculate $2^4$ by multiplying 2 by itself 4 times: 2 × 2 × 2 × 2 = 16. For negative exponents, such as $x\; =\; -3$, rewrite as a reciprocal: $2^\{-3\}\; =\; rac\{1\}\{2^3\}\; =\; rac\{1\}\{8\}$. For non-integer or large exponents, use a calculator with the caret (^) key to compute the value accurately.

What are the restrictions on the base of an exponential function?

The base of an exponential function must be a constant number that is positive and not equal to 1. This means the base cannot be a variable or a negative number. For example, bases like 2, 3/2, or 10 are valid because they are constant and positive. However, a base of 1 is not allowed because $1^x$ always equals 1 regardless of $x$, which does not produce exponential growth or decay. Similarly, negative bases are excluded because they can produce complex or undefined values when raised to variable exponents.

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

To evaluate an exponential function with a non-integer exponent, such as $2^\{3.14\}$, use a scientific calculator. First, enter the base number (e.g., 2). Then use the caret key ($^$) to indicate the exponent. Next, enter the exponent value (e.g., 3.14). It is helpful to use parentheses around the exponent to ensure the calculator interprets it correctly, like $2^\{(3.14)\}$. The calculator will then provide the decimal approximation of the value, which in this case is approximately 8.815. This method works for any real number exponent.

Can the exponent in an exponential function be an expression, and how does that affect the function?

Yes, the exponent in an exponential function can be an expression involving variables and constants, such as $x\; +\; 1$ in $10^\{x+1\}$. This means the entire expression is the power to which the base is raised. It affects the function by shifting or scaling the growth or decay depending on the expression. For example, $10^\{x+1\}\; =\; 10^x\; imes\; 10^1\; =\; 10\; imes\; 10^x$, which is a scaled version of $10^x$. Understanding the full exponent expression is important for evaluating and graphing the function accurately.