Identifying Exponential Functions With X-Intercepts A Comprehensive Guide

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Hey guys! Today, we're diving deep into the fascinating world of exponential functions and their x-intercepts. Let's crack this question together and make sure we understand the concepts inside out. Understanding x-intercepts is crucial for grasping the behavior and applications of exponential functions, so buckle up and let’s get started!

What are Exponential Functions?

Before we jump into the question, let's quickly recap what exponential functions are all about. An exponential function is a function of the form f(x) = a^x + k, where a is a constant greater than 0 and not equal to 1, and k is a constant that determines the vertical shift of the graph. The variable x appears as an exponent, which gives these functions their unique characteristics. For instance, consider f(x) = 2^x. As x increases, f(x) increases exponentially, making these functions essential in modeling growth and decay phenomena.

Key Characteristics of Exponential Functions

Exponential functions have several distinguishing features that set them apart from other types of functions:

  1. Horizontal Asymptote: A horizontal asymptote is a horizontal line that the graph of the function approaches as x tends to positive or negative infinity. The basic exponential function f(x) = a^x has a horizontal asymptote at y = 0. Transformations such as vertical shifts (adding or subtracting a constant) can move this asymptote up or down. For example, in f(x) = a^x + k, the horizontal asymptote is at y = k. Understanding the horizontal asymptote helps us predict the long-term behavior of the function. The function will get closer and closer to the asymptote but never actually cross it.

  2. Y-intercept: The y-intercept is the point where the graph of the function intersects the y-axis. It occurs when x = 0. For the basic exponential function f(x) = a^x, the y-intercept is always (0, 1) since any number (except 0) raised to the power of 0 is 1. For a transformed function f(x) = a^x + k, the y-intercept can be found by substituting x = 0 into the equation, giving f(0) = a^0 + k = 1 + k. The y-intercept is a critical point for understanding the initial value of a quantity being modeled by the exponential function.

  3. X-intercept: The x-intercept is the point where the graph of the function intersects the x-axis. It occurs when f(x) = 0. Not all exponential functions have x-intercepts. In fact, many do not. The existence of an x-intercept depends on the vertical shift and the base a. If the horizontal asymptote is above the x-axis (i.e., k > 0), the function will not have an x-intercept because it never crosses the x-axis. On the other hand, if the horizontal asymptote is below the x-axis (i.e., k < 0), the function may have an x-intercept.

  4. Monotonicity: Exponential functions are either strictly increasing or strictly decreasing, depending on the base a. If a > 1, the function is increasing, meaning that as x increases, f(x) also increases. If 0 < a < 1, the function is decreasing, meaning that as x increases, f(x) decreases. This property is particularly useful in real-world applications, such as modeling population growth (increasing) or radioactive decay (decreasing).

The Significance of X-Intercepts

Now, why are x-intercepts so important? The x-intercept of a function is the point where the function's value is zero. In the context of exponential functions, finding the x-intercept helps us determine when the quantity being modeled reaches zero. For example, if we're modeling the decay of a radioactive substance, the x-intercept (if it exists) would tell us when the substance has completely decayed. However, it’s crucial to remember that not all exponential functions have x-intercepts. This usually happens when the function has been shifted vertically upwards, so it never crosses the x-axis. Exponential functions are widely used in various fields like finance, biology, and physics, making understanding their intercepts incredibly valuable.

Breaking Down the Question

The question asks: Which exponential function has an x-intercept?

To find out which function has an x-intercept, we need to determine which function equals zero for some value of x. This means we're looking for a function f(x) such that f(x) = 0 has a real solution for x. Remember, an x-intercept occurs where the graph of the function crosses the x-axis, which is where y = 0.

Let's examine each option closely:

A. f(x) = 100^(x-5) - 1 B. f(x) = 3^(x-4) + 2 C. f(x) = 7^(x-1) + 1 D. f(x) = -8^(x+1) - 3

Our goal is to find which of these equations can be set to zero and solved for x. We'll look at each option individually and apply our knowledge of exponential functions to determine if an x-intercept exists.

Analyzing the Options

Let's dive into each option one by one. We'll set each function equal to zero and see if we can solve for x. If we can find a real value for x that makes the function equal to zero, then that function has an x-intercept. If not, it doesn't.

Option A: f(x) = 100^(x-5) - 1

To find the x-intercept, we set f(x) = 0:

100^(x-5) - 1 = 0

Now, let’s solve for x:

100^(x-5) = 1

Since any non-zero number raised to the power of 0 is 1, we can write:

x - 5 = 0

Solving for x:

x = 5

So, for option A, we found an x-intercept at x = 5. This means that the graph of f(x) = 100^(x-5) - 1 crosses the x-axis at the point (5, 0). We've already found a potential answer, but let's analyze the other options to make sure we have the correct one.

Option B: f(x) = 3^(x-4) + 2

Again, we set f(x) = 0:

3^(x-4) + 2 = 0

Now, let’s try to solve for x:

3^(x-4) = -2

Here’s where things get tricky. Exponential functions with a positive base (like 3) always produce positive results. There’s no real number x that we can plug in to make 3^(x-4) equal to a negative number like -2. Therefore, there is no x-intercept for this function. The graph of f(x) = 3^(x-4) + 2 never crosses the x-axis.

Option C: f(x) = 7^(x-1) + 1

Set f(x) = 0:

7^(x-1) + 1 = 0

Solve for x:

7^(x-1) = -1

Similar to option B, this equation has no real solution. An exponential function with a positive base (7 in this case) will always result in a positive value. It can never be equal to -1. So, this function also does not have an x-intercept. The graph of f(x) = 7^(x-1) + 1 remains above the x-axis.

Option D: f(x) = -8^(x+1) - 3

Set f(x) = 0:

-8^(x+1) - 3 = 0

Solve for x:

-8^(x+1) = 3

-8^(x+1) = 3 implies 8^(x+1) = -3

Again, we run into the same issue. While the base is negative in this case, it's the term 8^(x+1) that we need to consider. Exponential functions with a positive base (which 8 is) will always yield a positive result. Therefore, 8^(x+1) can never equal -3, and this function has no x-intercept. The graph of f(x) = -8^(x+1) - 3 does not cross the x-axis.

The Verdict

After analyzing all the options, we found that only option A, f(x) = 100^(x-5) - 1, has an x-intercept. We were able to solve the equation 100^(x-5) - 1 = 0 and find that x = 5. Options B, C, and D did not have real solutions when set to zero, meaning their graphs do not cross the x-axis.

Final Answer

Therefore, the correct answer is:

A. f(x) = 100^(x-5) - 1

Key Takeaways

  1. An x-intercept exists for an exponential function if setting the function equal to zero yields a real solution for x.
  2. Exponential functions of the form a^x, where a > 0, are always positive. Adding a constant k shifts the graph vertically, and the presence of an x-intercept depends on whether this shift brings the graph across the x-axis.
  3. If you encounter an equation where a^x equals a negative number, there is no real solution, and the function has no x-intercept.
  4. Understanding the behavior and transformations of exponential functions is crucial for determining their x-intercepts.

I hope this detailed explanation helps you understand how to find x-intercepts of exponential functions. Keep practicing, and you'll master these concepts in no time! If you have any more questions, feel free to ask. Keep up the great work, guys!