Why is the asymptote of the function in Example #1 y=2 ? ( y = 4x+3 + 2 )

the question is not complete..

linear functions do not have asymptote...

To determine why the asymptote of the function y = 4x+3 + 2 is y = 2, we need to analyze the characteristics of the given equation. Specifically, we will focus on the concept of asymptotes and how they relate to linear functions.

First, let's clarify the equation. The given function is y = 4x+3 + 2. This equation represents a linear function with a slope of 4 and a y-intercept of 5 (since 3 + 2 = 5).

To find the asymptote, we need to examine the behavior of the function as x approaches positive or negative infinity.

For a linear function, the general form is y = mx + b, where m represents the slope and b represents the y-intercept. The slope, in this case, is 4, which means that the function y = 4x+3 + 2 has a constant rate of change of 4. This implies that as x increases or decreases, y will also increase or decrease by a factor of 4.

Now, let's consider what happens as x approaches infinity (positive or negative). When x becomes significantly large (tending to infinity), the constant term 2 becomes insignificant compared to the larger values of 4x+3. Consequently, the equation simplifies to y ≈ 4x+3.

In this simplified form, it is evident that there is no horizontal line (asymptote) that the function approaches as x tends to infinity. The value of y keeps increasing indefinitely as x increases without bound. Therefore, we can conclude that there is no horizontal asymptote for the given function as x approaches infinity.

However, if we consider the behavior of the function as x approaches negative infinity, we can observe that y will keep decreasing indefinitely. In this case, the function does approach a horizontal line as x tends towards negative infinity. The value that the function approaches is y = 2, which serves as the horizontal asymptote in this scenario.

In summary, the asymptote of the function y = 4x+3 + 2 is y = 2 because as x approaches negative infinity, the function approaches this horizontal line.