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

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 if x are not allowed since x > 0 is specified.

Determine the behaviour 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 as horizantal asymptote at y = 3.

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.

Match the behaviour with options:
→ The function is not strictly linear because of the term \$1/"x"\$.
→ Its not strictly exponential because it doesn’t grow at an exponential rate.
→ But the behaviour is closest to that of anincreasing exponential function because
it approaches a horizontal asymptote as x increases.

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

Decreasing exponential

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

Decreasing linear

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

Increasing exponential

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

Increasing linear

Option

C

Can you summarize what you’ve understood in the above steps?