Question

to convert from (x) degrees in celsius to (y) degrees in fahrenheit, we use the formula (f(x) = \frac{9}{5}x + 32), which is an increasing function.

which of the following gives a verbal representation of the inverse function, (f^{-1}(x))?

• (f^{-1}) is a decreasing function. as the (x)-values increase, the (y)-values decrease.
• (f^{-1}) is a decreasing function. as the (x)-values decrease, the (y)-values decrease.
• (f^{-1}) is an increasing function. as the (x)-values increase, the (y)-values increase.
• (f^{-1}) is an increasing function. as the (x)-values increase, the (y)-values remain constant.

Explanation:

⚡ Using what you learned: inverse functions: definition and notation

Step 1: Analyze the behavior of the original function

The original function is:

$$ f(x) = \frac{9}{5}x + 32 $$

We are given that \( f(x) \) is an increasing function. This means as the input \( x \) increases, the output \( y \) also increases.

Step 2: Determine the behavior of the inverse function

An inverse function \( f^{-1}(x) \) reverses the inputs and outputs of the original function, but it preserves the direction of the relationship (monotonicity):

• If a function is strictly increasing, its inverse is also strictly increasing.
• Therefore, \( f^{-1} \) is an increasing function.

Step 3: Match with the verbal definition of an increasing function

By definition, for an increasing function:

• As the \( x \)-values increase, the \( y \)-values must also increase.

This matches the third option:
"\( f^{-1} \) is an increasing function. As the \( x \)-values increase, the \( y \)-values increase."

Answer:

\( f^{-1} \) is an increasing function. As the \( x \)-values increase, the \( y \)-values increase.