Exponential Functions: Videos & Practice Problems

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 and cannot equal 1, while the exponent must contain a variable.

For instance, consider the function f(x) = (2/3)^x. Here, the base is 2/3, which is positive and not equal to 1, confirming that this is indeed an exponential function. The exponent is simply x. Conversely, the function f(y) = 1^y does not qualify as an exponential function because the base is 1, which violates the criteria. Another example, f(x) = 10^(x+1), is valid since the base (10) is positive and not equal to 1, and the exponent x + 1 contains a variable.

Evaluating exponential functions involves substituting values for the variable in the exponent. For example, to evaluate f(x) = 2^x at x = 4, you compute 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, a calculator is often necessary. You would input 2, use the caret key for exponentiation, and then enter 3.14 to find that f(3.14) ≈ 8.815.

Lastly, for larger exponents, such as x = 12, you can use a calculator to compute f(12) = 2^{12} = 4096. This approach simplifies the process of evaluating exponential functions, especially when the exponent is large.

Graphing exponential functions, such as \( f(x) = 2^x \), is a straightforward process compared to polynomial and rational functions. To begin, we can generate ordered pairs by substituting values for \( x \). For instance, when \( x = 0 \), \( f(0) = 2^0 = 1 \). This gives us our first point at (0, 1). Continuing with \( x = 1 \), we find \( f(1) = 2^1 = 2 \), leading to the point (1, 2). As \( x \) increases, the function grows rapidly: \( f(2) = 4 \), \( f(3) = 8 \), and \( f(10) = 1024 \). Thus, we observe that as \( x \) approaches infinity, \( f(x) \) also approaches infinity, indicating that the end behavior on the right side of the graph is upward.

For negative values of \( x \), we see a different trend. For \( x = -1 \), \( f(-1) = 2^{-1} = \frac{1}{2} \), resulting in the point (-1, 0.5). As \( x \) becomes more negative, the function values decrease, approaching 0 but never actually reaching it. This behavior indicates the presence of a horizontal asymptote at \( y = 0 \), which we can represent with a dashed line. Therefore, as \( x \) approaches negative infinity, \( f(x) \) approaches 0.

Exponential functions are continuous, meaning there are no breaks in the graph. Additionally, they exhibit a one-to-one property, which means that each \( x \) value corresponds to a unique \( y \) value, allowing the graph to pass the horizontal line test. The overall shape of the graph resembles a stretched lowercase 'e', which can help in visualizing its form.

When analyzing the domain and range of exponential functions, the domain is always all real numbers, represented as \( (-\infty, \infty) \). The range, however, depends on the asymptote. For \( f(x) = 2^x \), the range is \( (0, \infty) \), as the function never touches the asymptote at \( y = 0 \).

Exploring other exponential functions of the form \( f(x) = b^x \), we note that the base \( b \) significantly influences the graph's behavior. If \( b > 1 \), the graph will always increase, while if \( 0 < b < 1 \), the graph will decrease. The steepness of the graph also varies with the base: for bases greater than 1, larger values of \( b \) yield steeper graphs, while for bases between 0 and 1, smaller values of \( b \) result in steeper graphs. This understanding of the relationship between the base and the graph's characteristics is crucial for effectively graphing exponential functions.

To graph the function \( f(x) = \left(\frac{1}{2}\right)^x \), we start by evaluating the function at various values of \( x \). This process involves substituting different \( x \) values into the function to find corresponding \( y \) values, which we can then plot on a graph.

First, when \( x = 0 \), we calculate:

\( f(0) = \left(\frac{1}{2}\right)^0 = 1 \)

This gives us the point \( (0, 1) \).

Next, for \( x = 1 \):

\( f(1) = \left(\frac{1}{2}\right)^1 = \frac{1}{2} \)

We plot the point \( (1, \frac{1}{2}) \).

For \( x = 2 \):

\( f(2) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \)

This gives us the point \( (2, \frac{1}{4}) \).

Continuing with \( x = 3 \):

\( f(3) = \left(\frac{1}{2}\right)^3 = \frac{1}{8} \)

We plot the point \( (3, \frac{1}{8}) \).

Now, we explore negative values of \( x \). Starting with \( x = -1 \):

\( f(-1) = \left(\frac{1}{2}\right)^{-1} = 2 \)

This gives us the point \( (-1, 2) \).

For \( x = -2 \):

\( f(-2) = \left(\frac{1}{2}\right)^{-2} = 4 \)

We plot the point \( (-2, 4) \).

Finally, for \( x = -3 \):

\( f(-3) = \left(\frac{1}{2}\right)^{-3} = 8 \)

This gives us the point \( (-3, 8) \).

As we analyze the points plotted, we observe that as \( x \) increases, the \( y \) values decrease, approaching zero but never actually reaching it. This behavior indicates the presence of a horizontal asymptote at \( y = 0 \). The graph will approach this line but will not touch it, which we denote with a dashed line.

Additionally, it is important to note that the function \( \left(\frac{1}{2}\right)^x \) can be rewritten as \( 2^{-x} \). This transformation shows that the graph of \( f(x) = \left(\frac{1}{2}\right)^x \) is a reflection of the graph of \( f(x) = 2^x \) across the \( y \)-axis.

In terms of the domain and range of the function, the domain of all exponential functions is all real numbers, expressed as \( (-\infty, \infty) \). The range, determined by the asymptote, is from \( 0 \) to \( \infty \), written as \( (0, \infty) \), indicating that \( 0 \) is not included in the range.

In summary, the graph of \( f(x) = \left(\frac{1}{2}\right)^x \) exhibits exponential decay, characterized by a horizontal asymptote at \( y = 0 \), a domain of all real numbers, and a range from \( 0 \) to infinity.

When dealing with exponential functions, you may encounter the base e, which is approximately equal to 2.71828. Unlike variables, e is a constant, similar to the well-known number π. This means you can evaluate and graph functions with base e just like any other exponential function.

For example, consider the function f(x) = e^x. To evaluate this function for specific values of x, such as x = 2 or x = -3, you can use a scientific calculator. To find f(2), you would compute e^2 by using the second ln function on your calculator, which allows you to input the exponent. This results in approximately 7.39 when rounded to two decimal places. Similarly, for f(-3), you would calculate e^{-3}, which can also be expressed as 1/e^3. This yields approximately 0.05 when rounded.

The graph of f(x) = e^x shares the same general shape as other exponential functions, such as 2^x and 3^x. Since e is between 2 and 3, the graph of e^x will lie between the graphs of 2^x and 3^x. For more complex functions involving e, you can apply transformations similar to those used for other bases.

Understanding the significance of e is crucial, as it arises from the concept of continuous compounding in finance. The formula for compound interest approaches e as the number of compounding periods approaches infinity. Beyond finance, e is also essential in fields such as population growth modeling and radioactive decay, making it a vital constant in various scientific applications.

In summary, e is a fundamental number in mathematics that behaves like any other base in exponential functions, allowing for straightforward evaluation and graphing. Its applications in real-world scenarios highlight its importance in both theoretical and practical contexts.