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Chapter 1: Problem 28

Explain why \(F=\frac{9}{5} C+32\) represents a function.

Short Answer

Expert verified
The formula \(F=\frac{9}{5} C+32\) is a function because for each input (Celsius temperature), it produces exactly one output (Fahrenheit temperature).

Step by step solution

01

Define the Function

The formula given is \(F=\frac{9}{5} C+32\), which is a function that maps the temperature in degree Celsius (C) to degree Fahrenheit (F).
02

Verify Each Input Has Exactly One Output

We can verify that this formula is indeed a function by checking that for each given value of C, there is exactly one value of F. This is true because for any specific value of C that you input into the formula, you'll always get one and only one output value of F. This formula doesn't have any variables other than C in it, meaning for each C, there can only be one F.
03

Conclusion

Therefore, \(F=\frac{9}{5} C+32\) is a function since each input has exactly one output. It fulfills the definition of a function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Temperature Conversion
Temperature conversion involves changing a temperature reading from one scale to another. The most common scales are Celsius (掳C) and Fahrenheit (掳F).
These scales are based on different fixed points and increments. The formula for converting Celsius to Fahrenheit is \( F = \frac{9}{5}C + 32 \). This formula works because:
  • The coefficient \( \frac{9}{5} \) adjusts for the difference in the size of the degrees between the Celsius and Fahrenheit scales.
  • The addition of 32 accounts for the different starting points of these scales, where water freezes at 0掳C and 32掳F.
By using this formula, you can easily switch between these temperature measures, aiding in interpreting weather forecasts, scientific data, and everyday activities that require temperature readings.
Linear Equations
Linear equations are mathematical expressions that model relationships between variables with a constant rate of change. They appear as straight lines when graphed. In the context of temperature conversion, \( F = \frac{9}{5} C + 32 \) is a linear equation.
This equation is linear because the highest power of the variable \( C \) is 1.
The main features of this linear equation include:
  • A constant rate of change, shown by the coefficient \( \frac{9}{5} \), indicating how much \( F \) changes when \( C \) changes.
  • An intercept, here the constant 32, showing where the line crosses the y-axis in a graph where \( C = 0 \).
  • The equation represents a direct proportional relationship between \( C \) and \( F \).
Understanding linear equations helps in predicting one variable based on another, which is vital in fields like physics, engineering, and economics.
One-to-One Correspondence
A one-to-one correspondence in mathematics ensures each input has a unique output. This principle is key to defining a function. With the formula \( F = \frac{9}{5} C + 32 \), each Celsius temperature maps to one specific Fahrenheit temperature.
Ensuring a one-to-one relationship is crucial in temperature conversion because:
  • There should be no ambiguity: no two different inputs should yield the same output.
  • Each input of Celsius must provide a distinct output of Fahrenheit to correctly display temperature changes.
  • This property means you can always reverse the process and convert Fahrenheit back to Celsius, maintaining correctness.
Hence, this function illustrates how every Celsius reading corresponds to a unique and predictable Fahrenheit value, making temperature conversions consistent and reliable.

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