Question text Consider the exponential function [math: f(x)=25x−4+4]\displaystyle f(x)={2^{5\,x-4}+4}. Determine the equation of the horizontal asymptote of [math: y=f(x)]. Enter your answer as [math: y=a], where [math: a] is an integer. [input] Check Question 7

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BlackTom AI · Accepted Answer

We start by identifying the form of the function given. The expression is an exponential function of the form f(x) = 2^{5x - 4} + 4. In exponential functions, the horizontal asymptote is determined by the behavior of the exponential term as x grows very large in either the positive or negative direction.

Consider the end behavior as x → -∞. The exponent 5x - 4 becomes a very large negative number, so 2^{5x - 4} approaches 0. Consequently, f(x) ≈ 0 + 4 in this limit, which means the function values approach 4.

Therefore, the horizontal line that the graph approaches as x decreases without bound is y = 4. This is the horizontal asymptote, since the exponential term vanishes and the constant term dominates the end behavior.

If someone were to misinterpret the function or its end behavior, they might think the asymptote is at y = 0 (ignoring the +4), or confuse the role of the exponent by focusing on growth as x → ∞ (where 2^{5x - 4} grows without bound, but that does not affect the horizontal asymptote in the negative direction). However, the standard definition of a horizontal asymptote concerns the behavior as x → ±∞, and in this case the relevant limit as x → -∞ yields y = 4.

Thus, by analyzing the limit of the non-constant part and its impact on f(x) for very negative x, we arrive at the asymptotic value of 4 for the function.

MCD2130 - T2 - 2025 MCD2130 Sample Test 3

Question text Consider the exponential function [math: f(x)=25x−4+4]\displaystyle f(x)={2^{5\,x-4}+4}. Determine the equation of the horizontal asymptote of [math: y=f(x)]. Enter your answer as [math: y=a], where [math: a] is an integer. [input] Check Question 7

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