Final Answer:
The end behavior of the function h(x) = eˣ + 1 is that it will approach positive infinity as x approaches positive infinity and will approach 1 as x approaches negative infinity.
Step-by-step explanation:
The end behavior of a function refers to the behavior of the function as the input, x, approaches infinity or negative infinity. To determine the end behavior of the function h(x) = eˣ + 1, we'll consider two separate cases: as x approaches positive infinity and as x approaches negative infinity.
**As x approaches positive infinity (x → ∞):**
- The term eˣ grows without bound because the exponential function increases at an ever-increasing rate as x gets larger.
- Therefore, as x approaches positive infinity, eˣ also approaches positive infinity.
- Since h(x) is just eˣ plus 1, the +1 has no effect on the end behavior as x approaches positive infinity.
- Hence, as x approaches positive infinity, h(x) also approaches positive infinity.
In mathematical notation, this is expressed as:
lim(x→∞) h(x) = lim(x→∞) (eˣ + 1) = ∞
**As x approaches negative infinity (x → -∞):**
- The term eˣ approaches zero because the exponential function rapidly decreases towards zero as x gets more negative.
- eˣ will become vanishingly small, so much so that its impact on the value of h(x) becomes negligible compared to the constant term +1.
- Therefore, as x approaches negative infinity, eˣ approaches 0, and h(x) approaches the constant term, which is 1.
In mathematical notation, this is written as:
lim(x→-∞) h(x) = lim(x→-∞) (eˣ + 1) = 1
**Summary of End Behavior:**
- As x approaches positive infinity, h(x) approaches positive infinity.
- As x approaches negative infinity, h(x) approaches 1.
This tells us that the graph of h(x) = eˣ + 1 will rise without limit as x gets larger and larger in the positive direction, and it will approach the horizontal line y = 1 as x decreases without bound in the negative direction.