Problem 2 - Algebra and Trigonometry

Question

By definition, \(f \circ g(x)=______\quad\) So if \(g(2)=5\) and \(f(5)=12,\) then \(f \circ g(2)= _______\)

Step-by-Step Solution

Answer

Therefore, \(f \circ g(2) = 12\).

Step 1: Understanding Composition of Functions

The composition of functions \(f \circ g\) means that first you apply function \(g\) to the input, and then apply function \(f\) to the result of \(g\). Mathematically, \(f \circ g(x) = f(g(x))\).

Step 2: Applying Function \(g\)

To find \(f \circ g(2)\), first evaluate \(g(2)\). According to the information given, \(g(2) = 5\).

Step 3: Applying Function \(f\)

Now, use the result from applying \(g\) as the input for \(f\), which means you need to evaluate \(f(5)\). According to the information given, \(f(5) = 12\).

Key Concepts

Composite Functions

Composite functions involve two different functions, which are combined to form a new function. The idea is to take the output from the first function and use it as the input for the second function.
When we denote a composite function as \(f \circ g(x)\), it means we will first apply function \(g\) to \(x\), and then apply function \(f\) to whatever result \(g(x)\) gives us.

• Start with the innermost function, \(g\), and evaluate it for the given input.
• Take the result of \(g(x)\) and use it as an input to the function \(f\).

This method of combining functions is very powerful, as it allows us to build more complex operations from simpler ones.

Evaluating Functions

Evaluating a function means finding the output for a specified input. If you have a function description, like \(g(x)\), you can plug in the given value of \(x\) to get the result.
For instance, if we are told \(g(2) = 5\), it means when \(x\) is 2, the function \(g\) outputs 5.

• Determine which value needs to be used as input.
• Substitute that input into the function.
• Perform any calculations required by the function to find the output.

In the context of composite functions, the process is almost like a relay race, where you get a result from one function and "pass it on" to the next function in line.

Ordered Pairs

In mathematics, ordered pairs are used to denote a pair of elements in a specific order, often represented as \((x, y)\).
These pairs are typically used to show a relationship between two quantities, like in graphs or mapping functions.
In the case of functions, the first element \(x\) represents the input, and the second element \(y\) is the output produced by the function when \(x\) is used as the input.

• The order of the elements is crucial; \((x, y)\) is not the same as \((y, x)\).
• These pairs help in understanding how inputs are mapped to outputs.

When dealing with composite functions, ordered pairs are invaluable in tracking how an input \(x\) transitions to an ultimate output \(f(g(x))\). This structured approach clarifies the operation of composite functions, showing each step clearly.