Problem 56 - Algebra 2

Question

ACT/SAT What is the value of \(f(g(6))\) if \(f(x)=2 x+4\) and \(g(x)=x^{2}+5 ?\) A 38 B 43 C 86 D 261

Step-by-Step Solution

Answer

The value of \( f(g(6)) \) is 86, which corresponds to option C.

Step 1: Understand the Functions

We need to find \( f(g(6)) \). This means first evaluating the function \( g \) at 6, and then using that result as the input for the function \( f \).

Step 2: Evaluate g(x) at x = 6

The function \( g(x) = x^2 + 5 \). Substitute 6 into \( g \): \[ g(6) = 6^2 + 5 = 36 + 5 = 41 \].

Step 3: Evaluate f(x) at the Result from g(x)

Now use the result from Step 2, where we found \( g(6) = 41 \), as the input for \( f(x) = 2x + 4 \). Calculate \( f(41) = 2(41) + 4 = 82 + 4 = 86 \).

Key Concepts

Function Composition

Function composition is an essential mathematical concept where one function is applied to the result of another. Consider it like fitting one process into another. For example, when we say we want to find \( f(g(x)) \), we first evaluate \( g(x) \) and then use its outcome as the input for \( f(x) \). This process is similar to inserting a key into a lock and turning it. The key (\( g(x) \)) unlocks \( f(x) \).

• Start with the inner function \( g(x) \).
• Find the value by substituting the given number.
• Take this result and plug it into the outer function \( f(x) \).

By following these steps, we can unravel what seems complex into a series of simple evaluations. In our exercise, first evaluate \( g(6) \), then insert that result into \( f(x) \).

Evaluating Functions

Evaluating a function means finding the output of the function for a given input. Functions like \( g(x) = x^2 + 5 \) and \( f(x) = 2x + 4 \) are rules that transform inputs into outputs. Consider \( g(x) \) and \( f(x) \) as machines:

• The input goes in (e.g., the number 6 for \( g(x) \)).
• The machine computes according to its rule (for \( g(x) \), squaring the input and adding 5).
• The output is then produced.

For \( g(6) \), substitute 6 into the machine and get 41. Next, input 41 into the machine \( f(x) \), and we output a final result of 86. Think of it as checking how the machine reacts to a particular input.

Function Notation

Function notation provides a compact and precise way of expressing functions. Instead of describing a function's operation with words, we use symbols. For example, \( f(x) = 2x + 4 \) tells us that \( x \), any number we choose, will be multiplied by 2 and then increased by 4 afterward. The letter \( f \) is just a symbol; it could be any letter and doesn't have inherent meaning except to differentiate between different functions.

• \( f(x) \) denotes that \( f \) is a function with \( x \) being the variable.
• \( g(x) \) does the same for another rule.
• It's a tidy way of differentiating operations.

Using function notation is like speaking a universal mathematical language. It helps us clearly define relationships between numbers, making complex equations, like composite functions, simpler to handle and understand.