1

Problem

For x > 0, the function h is defined as follows: \$h(x) = (1/x) + 3 \$. Which of the following could describe this function?

2

Step

Find the domain of the function:

→The function h(x) is defined for x > 0 because the expression \$1/x\$ is undefined when x = 0, and negative values of x are not allowed since it’s specified that x > 0.

3

Step

Determine the behavior as x approaches infinity:

→ As x approaches infinity, \$1/x\$ approaches 0.
→ So h(x) approaches 0 + 3 = 3 as x approaches infinity.
→ This indicates that the function has a horizontal asymptote at y = 3.

4

Step

Identify the trend of the function:

→ The term \$1/x\$ dominates the behaviour of the functionfor large values of x.
→ Since x is in the denominator, as x increases, \$1/x\$ decreases.
→ So, as x increases, h(x) increases and approaches y = 3.

5

Step

Match the behavior with options:

→ The function is not strictly linear due to the term \$1/x\$.
→ It’s not strictly exponential because it doesn’t grow at an exponential rate.
→ However, its behavior is closest to that of an increasing exponential function because it approaches a horizontal asymptote as x increases.

6

Sumup

Please summarize steps

Choices

7

Choice-A

This option is Incorrect because the function doesn’t exhibit exponential growth or decay

Wrong Decreasing exponential.

8

Choice-B

This option is Incorrect because the function overall increases due to the constant term

Wrong Decreasing linear.

9

Choice-C

This option is the Closest; while not strictly exponential, it describes the overall increasing trend towards a horizontal asymptote

Correct Increasing exponential.

10

Choice-D

This option is Incorrect because the function decreases as x increases due to the term \$1/x\$

Wrong Increasing linear.

11

Answer

Option

C

12

Sumup

Please summarize choices