Question

graph the function \\(f(x) = 12(1.3)^x\\), and use the graph to answer the following questions.

part a: describe the graph of \\(f(x)\\) using two of the following key features: y-intercept, asymptote, interval where the function is increasing or decreasing, and interval where the function is positive or negative. (10 points)

part b: what is the range of \\(f(x)\\)? use the graph to explain the answer. (4 points)

part c: what is the end behavior of \\(f(x)\\)? use the graph to explain the answer. (4 points)

Explanation:

Identify the function and key features

Using the Function Key Features and Intercepts of Functions knowledge points

$$
LATEXBLOCK0
$$

Determine the range of the function

Using the Function Key Features knowledge point

$$
LATEXBLOCK1
$$

Determine the end behavior

Using the End Behavior of Functions and Exponential End Behavior knowledge points

$$
LATEXBLOCK2
$$

Answer:

Part A

• y-intercept: The graph crosses the y-axis at \((0, 12)\).
• Asymptote: The graph has a horizontal asymptote at \(y = 0\).

Part B
The range of \(f(x)\) is \((0, \infty)\), or \(y > 0\). Since the horizontal asymptote is \(y = 0\) and the curve lies entirely above the x-axis, the function values are always strictly positive.

Part C

• As \(x \to \infty\), \(f(x) \to \infty\).
• As \(x \to -\infty\), \(f(x) \to 0\).