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, which is a variable. 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 is f(x) = 10^(x+1). In this case, the base is 10, which is positive and not equal to 1, while the exponent is x + 1, containing the variable x. Thus, this function is also 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, we compute f(4) = 2^4 = 16. When evaluating at negative values, such as x = -3, we find f(-3) = 2^{-3} = 1/(2^3) = 1/8.

For non-integer values, such as x = 3.14, a calculator is often necessary. To compute f(3.14) = 2^{3.14}, you would input 2, use the caret key for exponentiation, and then enter 3.14, yielding approximately 8.815.

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

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: \( f(2) = 4 \) and \( f(3) = 8 \). For larger values, such as \( x = 10 \), \( f(10) = 2^{10} = 1024 \). This indicates that as \( x \) approaches infinity, \( f(x) \) also approaches infinity, demonstrating the end behavior of the graph.

When considering negative values of \( x \), such as \( x = -1 \), we calculate \( f(-1) = 2^{-1} = \frac{1}{2} \), 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 can be represented by a dashed line on the graph. Thus, 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 implies that no two \( x \) values yield the same \( y \) value. This can be verified by the horizontal line test, where any horizontal line intersects the graph at most once.

The overall shape of the graph resembles a stretched lowercase 'e', which can help in visualizing its form. The domain of any exponential function is all real numbers, expressed as \( (-\infty, \infty) \). The range, however, is determined by the asymptote; for \( f(x) = 2^x \), the range is \( (0, \infty) \), indicating that \( y \) values are always positive and never include 0.

When exploring other exponential functions of the form \( f(x) = b^x \), 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. Conversely, for bases between 0 and 1, smaller values of \( b \) result in steeper graphs.

Understanding these characteristics of exponential functions allows for effective graphing and analysis, providing a solid foundation for further exploration in mathematics.

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 will evaluate 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 observe the values, we notice that for positive \( x \), the function yields fractions that approach zero, while for negative \( x \), the function yields whole numbers that increase. This behavior indicates that as \( x \) increases, \( f(x) \) approaches the x-axis but never touches it, suggesting a horizontal asymptote at \( y = 0 \).

It is also important to note that \( f(x) = \left(\frac{1}{2}\right)^x \) can be rewritten as \( f(x) = 2^{-x} \). This transformation shows that the graph of \( f(x) \) is a reflection of the graph of \( g(x) = 2^x \) 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 horizontal asymptote, is \( (0, \infty) \), indicating that the function values are always positive and never reach zero.

In summary, the graph of \( f(x) = \left(\frac{1}{2}\right)^x \) features a horizontal asymptote at \( y = 0 \), a domain of all real numbers, and a range from \( 0 \) to infinity. Understanding these characteristics is essential for accurately graphing and interpreting exponential functions.

When exploring 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 involving e just like any other exponential function.

For instance, 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. For f(2), you would compute e^2 by using the second ln function on your calculator, which yields 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 results in 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 these two functions. For more complex functions involving e, you can apply transformations similar to those used for other bases.

Understanding the significance of e is crucial, particularly in contexts like compound interest. The formula for compound interest approaches e as the number of compounding periods increases indefinitely. This concept extends beyond finance; e is also integral in modeling population growth, radioactive decay, and various natural phenomena.

In summary, e is a fundamental constant in mathematics that simplifies the study of exponential growth and decay across multiple disciplines. By treating it like any other number, you can effectively evaluate and graph functions involving e, enhancing your understanding of exponential behavior in real-world applications.