Problem 82 - Precalculus

Question

Find the \(y\) -intercept and horizontal asymptote(s) of \(f(x)=2^{x}+3^{x}\)

Step-by-Step Solution

Answer

The y-intercept is (0, 2) and there is no horizontal asymptote.

Step 1: Understanding the y-Intercept

The y-intercept of a function is the point where the graph intersects the y-axis. This occurs when the input value, or x-value, is zero. To find the y-intercept of the function, substitute 0 for x in the function equation: \\(f(0) = 2^0 + 3^0\).\Evaluate the expression, \\(f(0) = 1 + 1 = 2\).\So, the y-intercept is (0, 2).

Step 2: Exploring Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a graph as x approaches infinity or negative infinity. For the function \(f(x) = 2^x + 3^x\), as \(x\) approaches infinity, both \(2^x\) and \(3^x\) grow exponentially, with \(3^x\) dominating because its base is larger. Thus, the function tends towards \(f(x) \approx 3^x\) as \(x\to\infty\). Therefore, there is no horizontal asymptote because \(3^x\) does not level off at a constant value.

Key Concepts

Understanding the y-Intercept

In precalculus, the y-intercept is a fundamental point on the graph of a function. The y-intercept occurs where the graph crosses the y-axis, meaning it has an x-coordinate of zero. Essentially, it's the point \(0, f(0)\). Finding this point involves substituting 0 into the function for x.
For example, consider the function \(f(x) = 2^x + 3^x\). To find the y-intercept:

• Substitute 0 for x: \(f(0) = 2^0 + 3^0\).
• Calculate the expression: \(2^0 = 1\) and \(3^0 = 1\).
• Therefore, \(f(0) = 1 + 1 = 2\).
• The y-intercept is then the point (0, 2).

This procedure enables you to quickly identify where a graph hits the y-axis.

Exploring Horizontal Asymptotes

Horizontal asymptotes give insight into the end-behavior of a function. They describe how a function behaves as x moves towards infinity (or negative infinity). In cases where a function flattens out, you'll usually find a horizontal asymptote, which is a constant horizontal line.
Consider the function \(f(x) = 2^x + 3^x\). As x approaches infinity, both \(2^x\) and \(3^x\) grow exponentially larger, but \(3^x\) grows faster due to its larger base. Consequently, the function becomes dominated by \(3^x\), making \(f(x) \approx 3^x\) for very large x-values.

• This behavior implies a direction rather than flattening to a single horizontal line.
• Thus, in this particular case, no horizontal asymptote exists.

Understanding this helps you grasp why some functions never stabilize to a horizontal line as they go infinitely in one direction.

Exponential Functions

Exponential functions are a staple in precalculus and beyond. An exponential function typically takes the form \(a^x\), where a is a positive constant. Such functions are characterized by their rapid rate of increase or decrease. Here's what defines them:

• The base a is greater than 1. If a is less than 1, the function decreases.
• Growth is multiplicative and accelerates quickly past small values of x.
• They never hit zero, as \(a^x\) is always positive.

In \(f(x) = 2^x + 3^x\), the presence of two exponential terms indicates even faster growth.
This is because each component contributes to the overall increase without limiting the other. Familiarity with these properties aids in predicting graph behaviors and solving complex equations.