Problem 35 - Algebra 2

Question

Let \(f(x)=x^{2}\) and \(g(x)=x-3 .\) Find each value or expression. $$ (g \circ f)(3.5) $$

Step-by-Step Solution

Answer

The value of \((g \circ f)(3.5)\) is 9.25.

Step 1: Understand the Functions

We have two functions: \(f(x)=x^{2}\) and \(g(x)=x-3\). When we see \(g \circ f(x)\), it means we first apply the function \(f\) to \(x\), and then apply \(g\) to the result.

Step 2: Apply the first function

We first apply the function \(f\) to our input 3.5. This means we substitute 3.5 in for \(x\) in the function \(f(x)\), so we get \(f(3.5)=(3.5)^{2}=12.25\).

Step 3: Apply the second function

Next we take the result from Step 2, which is 12.25, and put that into the function \(g\). We substitute 12.25 for \(x\) in \(g(x)\), resulting in \(g(12.25)=12.25-3=9.25\).

Key Concepts

Functions

Functions are fundamental elements used in mathematics to map or transform inputs into outputs. Think of a function as a rule or a machine that takes an input and provides an output based on a specific formula. For example, in the given exercise, we have two different functions:

• Function 1: \(f(x) = x^2\) - This is a quadratic function that squares its input. So, if you input 3, the output will be \(3^2 = 9\).
• Function 2: \(g(x) = x - 3\) - This function takes an input and subtracts 3 from it. For example, if the input is 5, the output will be \(5 - 3 = 2\).

Functions can be visualized as processes that change or transform numbers. Each function's definition provides a clear map of how any input is transformed through its formula.

Function Composition

Function composition is an operation that takes two functions and combines them to form a new function. The idea is to first apply one function and then apply the second one to the result. This is noted as \((g \circ f)(x)\), meaning we first use the function \(f\) and then \(g\). Here’s a simple way to understand it:

• First, apply \(f\) to \(x\), transforming it according to \(f(x)\).
• Next, take the output from \(f(x)\) and plug it into \(g\).

In the example from the exercise, when we calculate \((g \circ f)(3.5)\), we start by finding \(f(3.5)\) and use the result in \(g\). We first transform 3.5 to 12.25 using \(f\), and then transform 12.25 to 9.25 using \(g\). This sequence of operations is what makes function composition such a powerful tool in mathematics.

Mathematical Operations

Mathematical operations are basic calculations that can involve addition, subtraction, multiplication, and division. These operations allow us to evaluate expressions and solve a wide variety of problems. In the context of functions, operations help us find new values by applying the formula of a function to given numbers or expressions. In the exercise:

• The first operation is squaring the input 3.5 using the function \(f\). This involves multiplication: \(3.5 \, \times \, 3.5\) resulting in 12.25.
• The second operation involves subtraction: taking the output from \(f(3.5)\), which is 12.25, and subtracting 3 as per the function \(g\). Thus, we calculate \(12.25 - 3 = 9.25\).

These operations are crucial in simplifying and evaluating the expressions that arise from function compositions. Understanding each step clearly—applying basic operations like squaring or subtracting—is essential for solving more complex function-based problems efficiently.