Question

The function for converting the temperature from degrees Fahrenheit, \(x,\) to degrees Celsius, \(y,\) is \(y=\frac{5}{9}(x-32)\) a) Determine the equivalent temperature in degrees Celsius for \(90^{\circ} \mathrm{F}\) b) Determine the inverse of this function. What does it represent? What do the variables represent? c) Determine the equivalent temperature in degrees Fahrenheit for \(32^{\circ} \mathrm{C}\) d) Graph both functions. What does the invariant point represent in this situation?

Short answer

a) 32.22°C. b) The inverse function is \( x = \frac{9}{5}y + 32 \), converting Celsius back to Fahrenheit. c) 32°C = 89.6°F. d) The invariant point is where the temperatures in Fahrenheit and Celsius are equal.

Step-by-step solution

Convert 90°F to Celsius

Use the given function to convert 90°F to Celsius. Substitute 90 for the variable x in the formula: \[ y = \frac{5}{9}(x-32) \] \[ y = \frac{5}{9}(90-32) \] \[ y = \frac{5}{9} \times 58 \] \[ y = 32.22 \] Therefore, 90°F is equivalent to 32.22°C.

Determine the inverse function

To find the inverse, solve for x in terms of y from the original function y = \frac{5}{9}(x-32): 1. Isolate x:\[ y = \frac{5}{9}(x-32) \] \[ y \times \frac{9}{5} = x - 32 \] \[ \frac{9}{5}y + 32 = x \] Thus, the inverse function is: \[ x = \frac{9}{5}y + 32 \]This represents converting from Celsius back to Fahrenheit, where y is the temperature in Celsius and x is the temperature in Fahrenheit.

Convert 32°C to Fahrenheit

Using the inverse function derived in Step 2, convert 32°C to Fahrenheit by substituting 32 for y: \[ x = \frac{9}{5}y + 32 \] \[ x = \frac{9}{5}(32) + 32 \] \[ x = 57.6 + 32 \] \[ x = 89.6 \] Therefore, 32°C is equivalent to 89.6°F.

Graph both functions

Graph the original function \( y = \frac{5}{9}(x-32) \) and its inverse \( x = \frac{9}{5}y + 32 \). Both graphs should be linear. The original function has a positive slope of \( \frac{5}{9} \), while the inverse has a slope of \( \frac{9}{5} \). The invariant point, where the two functions intersect, occurs when Fahrenheit equals Celsius. Solving for this, set \( y = x \) in either formula:\[ y = \frac{5}{9}(y-32) \] Solving this shows the point where F = C. This point is 40°C = 104°F.

Key concepts

function inverse

Understanding the inverse of a function is crucial in mathematics, especially when dealing with temperature conversions. The inverse function essentially reverses the original function. For instance, if our function converts Fahrenheit to Celsius, the inverse will convert Celsius back to Fahrenheit.

To find the inverse, start with the original function and solve for the original input variable. Let's take the function for converting Fahrenheit to Celsius:

\[ y = \frac{5}{9}(x - 32) \]

To make x the subject, follow these steps:

• Multiply both sides by \( \frac{9}{5} \) to isolate the term \(x-32\):\[ y \times \frac{9}{5} = x - 32 \]


• Add 32 to both sides: \[ \frac{9}{5}y + 32 = x \]

Therefore, the inverse function is: \[ x = \frac{9}{5} y + 32 \]

In this case, y represents the temperature in Celsius, and x represents the temperature in Fahrenheit. This inverse function allows you to switch easily between Celsius and Fahrenheit!

Celsius to Fahrenheit conversion

Temperature conversion is a common real-world application of linear functions. Converting from Fahrenheit to Celsius can be done using the formula:

\[ y = \frac{5}{9}(x - 32) \]

where x is the temperature in Fahrenheit and y is the temperature in Celsius. Let's convert 90°F to Celsius as an example:\

• Substitute 90 for x in the formula: \[ y = \frac{5}{9}(90 - 32) \]


• Simplify inside the parentheses: \[ y = \frac{5}{9}(58) \]


• Complete the multiplication to get: \[ y = 32.22 \]

Thus, 90°F is equivalent to 32.22°C.

Using the inverse function from earlier, you can convert from Celsius to Fahrenheit:

• Let's convert 32°C to Fahrenheit:


• Substitute 32 for y in the inverse function: \[ x = \frac{9}{5}(32) + 32 \]


• Simplify inside the parentheses: \[ x = 57.6 + 32 \]


• Complete the addition to get: \[ x = 89.6 \]

Therefore, 32°C is equivalent to 89.6°F. Understanding both formulas helps in accurately converting between the two temperature scales.

graphing linear functions

Graphing linear functions provides visual insight into the relationships between variables. For the temperature conversion functions, both the original and inverse functions are linear. To graph these, you'll plot their respective equations.

• The original function is \( y = \frac{5}{9}(x - 32) \), which has a slope of \( \frac{5}{9} \) and an x-intercept at 32. This means for every 9 units increase in x (Fahrenheit), y (Celsius) increases by 5 units.


• The inverse function is \( x = \frac{9}{5} y + 32 \), with a slope of \( \frac{9}{5} \). Here, for every 5 units increase in y, x increases by 9 units.

When you graph these, you'll notice they intersect at an invariant point where Fahrenheit equals Celsius.

Puzzle Moment: Does Fahrenheit ever equal Celsius? If you set \( y = x \) in any of the conversion formulas, you'll find that they intersect at -40°, the point at which both temperature scales are the same.

To summarize:
• The slope of the Fahrenheit to Celsius graph is \( \frac{5}{9} \).
• The slope of the Celsius to Fahrenheit graph is \( \frac{9}{5} \).


• The invariant point where F = C is -40°.
Graphing these lines helps you visualize how temperatures correlate between scales!