Question

Describe the end behavior, and intervals of increase/decrease for the exponential function graphed below.

The points on the graph are (0,1)


(5 points)
End Behavior:
As x approaches −∞
, f(x)
approaches Response area
As x approaches ∞
, f(x)
approaches Response area



Intervals of Increase/Decrease:

The exponential function is Response area from Response area to Response area.

answers

0

1

5

infinite sign

- infinite sign

increasing

decreasing

Answer 1

Let’s analyze the exponential function \( f(x) \) with the given point \( (0,1) \), which indicates that at \( x=0 \), \( f(x) = 1 \). Typically, a standard exponential function can be written in the form \( f(x) = a \cdot b^x \), where \( a > 0 \) and \( b > 1 \).

### End Behavior:
1. As \( x \) approaches \( -\infty \):
- For the exponential function, as \( x \) decreases indefinitely, \( f(x) \) approaches \( 0 \).
- Thus, **response area** for \( f(x) \) is **0**.

2. As \( x \) approaches \( \infty \):
- As \( x \) increases indefinitely, \( f(x) \) grows without bound.
- Thus, **response area** for \( f(x) \) is **infinity**.

### Intervals of Increase/Decrease:
- Exponential functions of the form \( a \cdot b^x \) (where \( b > 1 \)) are always increasing. Therefore:
- The exponential function is **increasing** on the interval **from \( -\infty \) to \( \infty \)**.

### Summary:

- End Behavior:
- As \( x \) approaches \( -\infty \), \( f(x) \) approaches **0**.
- As \( x \) approaches \( \infty \), \( f(x) \) approaches **infinity**.

- Intervals of Increase/Decrease:
- The exponential function is **increasing** from **\( -\infty \) to \( \infty \)**.