Question

If \(f(x)=2 x\) and \(g(x)=x+3\), what is the value of \(f(g(2))\) ? \(\begin{array}{lll}\text { (A) } & 10 & \text { OR }\end{array}\) (B) 13

Short answer

The value of \(f(g(2))\) is 10, so the answer is (A) 10.

Step-by-step solution

Understanding the Problem

We are given two functions, \(f(x) = 2x\) and \(g(x) = x+3\). We need to find the value of \(f(g(2))\). This means we first find \(g(2)\) and then substitute that result into the function \(f\).

Evaluate the Inner Function

First, evaluate the inner function \(g(x)\) at \(x = 2\). Substitute \(x = 2\) into \(g(x) = x + 3\).\[g(2) = 2 + 3 = 5.\]

Evaluate the Outer Function

Use the result from the inner function, \(g(2) = 5\), and substitute it into \(f(x) = 2x\):\[f(5) = 2(5) = 10.\]

Conclusion

The final result of \(f(g(2))\) is 10 based on the calculations above. Therefore, the answer is the value of \(f(g(2))\).

Key concepts

Understanding Mathematical Functions

Mathematical functions serve as a fundamental part of mathematics and are used to map an input known as the domain to an output, the range. Functions are expressed as equations, typically written in terms of a variable such as \( x \), where a rule or pattern dictates this relationship. In the context of our exercise, we have two functions: \( f(x) = 2x \) and \( g(x) = x + 3 \).Functions can be likened to machines:

• Input your number (here, 2 for the function \( g \)).
• Apply the rule ("add 3" for \( g(x) = x+3 \)).
• Receive the output ("5" in this case).

This concept helps deepen our understanding of how functions work and underscores the idea of mapping an input to a specific output, a core principle of mathematical functions.

The Role of Algebra in Functions

Algebra allows us to handle relationships between quantities using symbols and letters to denote numbers. It provides a way to express computations symbolically rather than always needing a specific number, such as the general formulas for \( f(x) \) and \( g(x) \) given in this exercise.

Function Composition in Algebra

When we compose functions, such as finding \( f(g(x)) \), we are essentially performing two algebraic tasks sequentially:

• First, calculate \( g(x) \), the output of the initial function.
• Then use that result as the new input for \( f(x) \).

Through algebra, we can visualize and work through this process systematically, enabling a clearer understanding and the ability to solve complex problems using function composition.

Solving Problems Using Function Composition

Problem solving with function composition involves a strategic approach to break down the task into simpler steps, thus reducing complexity. In the given problem, we need to find \( f(g(2)) \), which requires evaluating \( g \) at a particular point and then passing that result into \( f \).

Practical Problem Solving Steps

To solve problems involving function composition, follow these steps:

• **Identify:** Determine which functions and operations are needed and the order in which to perform them.
• **Compute the Inner Function:** Evaluate the first function with the given input. For example, for \( g(x) = x+3 \), we compute \( g(2) = 5 \).
• **Compute the Outer Function:** Use the result of the inner function as input for the next function, \( f(x) \). Thus, \( f(5) = 10 \).
• **Verify:** Always double-check the calculations to ensure accuracy, especially when dealing with nested functions.

Problem solving in mathematics benefits greatly from this structured approach, allowing for efficient and effective solutions to what might initially seem complex tasks.