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Fahrenheit Celsius Kelvin – composite functions

temperature

What is the composite function f o g?

Fahrenheit Celsius Kelvins
Fahrenheit Celsius Kelvins

f is the outer function; g is the inner function. `Suppose f(x) = x-2 and g(x) = x2. (a) Find fog and gof. (fog)(x) = f(g(x)) = f(x2) = x2-2.

In 1954, the tenth general conference on weights and measures adopted the Kelvin K as the basic unit for measuring all international weights and measures. While the kelvin is the standard unit, degrees Fahrenheit and degrees Celsius are still in common use in the United States.

3 temperature conversions
3 temperature conversions

The function C(F)=5/9(F-32) relates Celsius temperatures and Fahrenheit Temperatures. The function K(C)= C + 273.15 relates celsius temperatures and kelvin temperatures.

Let’s convert Fahrenheit to Kelvin. The composition of the function K with the function C is

K(C(F)) = C(F) + 273 = (5/9)(F-32) + 273

Since C converts Fahrenheit to Celsius and K converts Celsius to Kelvin, the composition will convert Fahrenheit to Kelvin.

In English:

  • To convert Fahrenheit to Celsius, subtract 32 from the Fahrenheit temperature and then divide your answer by 1.8.
  • To convert from Celsius to Fahrenheit, multiply the Celsius temperature by 1.8 and then add 32 to your answer.
  • If you’re trying to convert Celsius to Kelvin, just add 273.15 to the Celsius temperature.
At what temperature are Fahrenheit and Celsius the same?

Let’s solve a system of linear equations graphically then algebraically.

Graphing solution:

Celsius meets Fahrenheit
Celsius meets Fahrenheit

Algebraic solution:

The formulas for converting between degree Celsius and degree Fahrenheit are:

°F = (°C * 9/5) + 32
°C = (°F – 32) * 5/9

To find the temperature when both are equal, we use an old algebra trick and just set ºF = ºC and solve one of the equations.

Thermometers Fahrenheit Celsius Kelvin
Thermometers Fahrenheit Celsius Kelvin

°C = (°C * 9/5) + 32
°C – (°C * 9/5) = 32
-4/5 * °C = 32
°C = -32 * 5/4
°C = -40

°F = (°F * 9/5) + 32
°F – (°F * 9/5) = 32
-4/5 * °F = 32
°F = -32 * 5/4
°F = -40

So the temperature when both the Celsius and Fahrenheit scales are the same is -40 degrees.

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Classifications and Combinations of Functions

The modern notion of a function is derived from the efforts of many seventeenth- and eighteenth-century mathematicians. Of particular note was Leonhard Euler, to whom we are indebted for the function notation y = f(x). By the end of the eighteenth century, mathematicians and scientists had concluded that many real-world phenomena could be represented by mathematical models taken from a collection of functions called elementary functions.

Elementary functions fall into three categories.

  1. Algebraic functions (polynomial, radical, rational)
  2. Trigonometric functions (sine, cosine, tangent, and so on)
  3. Exponential and logarithmic functions

Source: Essential Calculus – Early Transcendental Functions by Ron Larson