Which of the following graphs represents f(x)=−2e−x+1 ?

Question

BlackTom AI · Accepted Answer

To identify which graph represents f(x) = -2e^{-x} + 1, I’ll break down the key features of this function first, then explain how they appear in a graph.

- Form and basic shape: The function is an exponential with a negative sign in front of the exponential term, plus a constant 1. This means the graph will be a translated exponential curve that increases as x grows, because e^{-x} decreases as x increases, and the negative sign makes -2e^{-x} increase toward 0 from negative values. Overall, the graph should rise from left to right.

- Horizontal asymptote: As x → ∞, e^{-x} → 0, so f(x) → 1. Therefore, the graph has a horizontal asymptote at y = 1.

- End behavior: As x → -∞, e^{-x} → ∞, so -2e^{-x} → -∞ and f(x) → -∞. This means the left end of the graph plunges downward without bound.

- Intercept: To find the x-intercept, set f(x) = 0: -2e^{-x} + 1 = 0 ⇒ e^{-x} = 1/2 ⇒ -x = ln(1/2) = -ln 2 ⇒ x = ln 2 ≈ 0.693. The graph crosses the x-axis at x ≈ 0.693, and y = 0 there.

- y-intercept: At x = 0, f(0) = -2e^{0} + 1 = -2 + 1 = -1. So the graph crosses the y-axis at (0, -1).

If you are choosing among multiple graphs, look for:
- A curve that is increasing from left to right and stays below y = 1, approaching y = 1 as x grows large.
- No vertical asymptote.
- Left end behavior trending downward to -∞.
- An x-intercept near x ≈ 0.69 and a y-intercept at (0, -1).

Any graph that matches these features (horizontal asymptote y = 1, increasing across the plane, left tail down to -∞, crosses x-axis near 0.693, crosses y-axis at -1) would be the correct representation of f(x) = -2e^{-x} + 1.

MATH-112-301-001 Unproctored Midcourse 1 Practice Exam 1

Which of the following graphs represents f(x)=−2e−x+1 ?

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