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Chapter 2: Problem 4

The function \(f(x)\) is defined as \(f(x)=2 g(x),\) where \(g(x)=x+5 .\) What is the value of \(f(3) ?\) \(\begin{array}{rr}{\text { (A) }} & {-4} \\ {\text { (B) }} & {6} \\ {\text { (C) }} & {8} \\ {\text { (D) }} & {16}\end{array}\)

Short Answer

Expert verified
The value of \( f(3) \) is 16 (D).

Step by step solution

01

- Understand the Functions

The function \( f(x) \) is defined as \( f(x) = 2g(x) \), where \( g(x) = x + 5 \).
02

- Evaluate \( g(3) \)

To find \( g(3) \), substitute \( x \) with 3 in the function \( g(x) = x + 5 \). So, \( g(3) = 3 + 5 = 8 \).
03

- Evaluate \( f(3) \)

Use the value of \( g(3) \) to find \( f(3) \). Since \( f(x) = 2g(x) \), substitute \( g(3) \) into this equation. So, \( f(3) = 2g(3) = 2(8) = 16 \).
04

- Select the Correct Answer

The value of \( f(3) \) is 16, which corresponds to option (D).

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Key Concepts

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

SAT math problem
The exercise provided is a classic type of SAT math problem.
SAT math problems often involve evaluating functions, algebraic expressions, and applying basic math operations.
This type of question tests your understanding of function definitions and compositions.
It's essential to carefully analyze the problem and follow a step-by-step approach to find the correct answer.
In this example, we are given two functions, and our goal is to evaluate them at a specific point.
defining functions
Defining functions is crucial in understanding how mathematical relationships work.
A function is essentially a rule that assigns each input exactly one output.
In this problem, we have two functions defined:
  • Function 1: f(x) = 2g(x)
  • Function 2: g(x) = x + 5
These definitions tell us precisely how to process any input value (x) to get the corresponding output.
For example, if we want to know the value of f(x) when x is any number, we first need to understand how g(x) transforms x.
After that, we can use the definition of f(x) to arrive at the final value.
function composition
Function composition involves applying one function to the results of another function.
In mathematical notation, it's often written as (f 鈭� g)(x).
For this problem, we are using compositions like f(x) and g(x).
Specifically, we see that f(x) builds on g(x).
To solve f(3), composition works in the following steps:
  • First, evaluate g(x) at the given point: g(3)
  • Next, use the result from g(3) in the definition of f(x) to find f(3).
This approach is helpful for breaking down complex problems into simpler steps.
By handling one function at a time, it becomes easier to avoid mistakes and understand each part of the problem.
evaluating functions
Evaluating functions is the process of calculating the output of a function given a specific input.
In our example, we need to find the value of f(3).
Here鈥檚 how we do it step by step:
  • First, determine g(3):
    If g(x) = x + 5, then g(3) = 3 + 5 = 8.
  • Next, use the evaluated value in the function f(x):
    Since f(x) = 2g(x), we substitute g(3) into the function:
    f(3) = 2g(3) = 2 * 8 = 16.
Thus, f(3) equals 16.
Having broken it down, 16 is the output when we input 3 into the function.
The correct answer is option (D).

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