Problem 30 - Intermediate Algebra

Question

Let \(f(x)=2 x-5\) and \(g(x)=x+1 .\) Find each of the following function values. See Example 2 . $$ (f-g)(-4) $$

Step-by-Step Solution

Answer

\((f-g)(-4) = -10\).

Step 1: Understand the Problem

We need to find the value of the function \((f-g)(x)\) evaluated at \(x = -4\). The function \((f-g)(x)\) represents the difference between the functions \(f(x)\) and \(g(x)\).

Step 2: Define (f-g)(x)

The difference of two functions \(f\) and \(g\) is defined as \((f-g)(x) = f(x) - g(x)\). Here, \(f(x) = 2x - 5\) and \(g(x) = x + 1\), so \((f-g)(x) = (2x - 5) - (x + 1)\).

Step 3: Simplify (f-g)(x)

Simplify the expression \((2x - 5) - (x + 1)\) by distributing the minus sign across \(g(x)\):\[(f-g)(x) = 2x - 5 - x - 1\]Now combine like terms:\[(f-g)(x) = (2x - x) + (-5 - 1) = x - 6\].

Step 4: Evaluate (f-g)(-4)

Substitute \(x = -4\) into the simplified expression for \((f-g)(x) = x - 6\):\[(f-g)(-4) = -4 - 6\].

Step 5: Calculate the Result

Calculate \(-4 - 6\):\((f-g)(-4) = -10\).

Key Concepts

Difference of Functions

In mathematics, one way to handle multiple functions is by considering the difference between them. When you hear about the "difference of functions," it simply means subtracting one function from another. This can be represented as

• \((f-g)(x) = f(x) - g(x)\)

For example, given two functions, \(f(x) = 2x - 5\) and \(g(x) = x + 1\), the difference is calculated by

• Rewriting it as \((f-g)(x) = (2x - 5) - (x + 1)\)

To break it down, you first tackle it by distributing the negative sign across each term in \(g(x)\). This helps in setting them up for simplification. The result is a new expression that is often much simpler and easier to work with. Understanding how to find the difference of functions is fundamental in calculus and algebra, aiding in the analysis of function behaviors.

Function Evaluation

Function evaluation involves plugging a specific value into a function and computing the result. Before you start, be sure you have a clear expression of the function. In our example, we are evaluating the expression for the difference of functions, which we found to be

• \((f-g)(x) = x - 6\)

To evaluate this at a specific number like \(-4\), simply substitute \(-4\) for each instance of \(x\) in the expression. Therefore, substituting gives you:

• \((f-g)(-4) = -4 - 6\)

This process doesn't end until you follow through with calculating the result. Function evaluation becomes particularly useful when comparing different segments of functions or determining specific outputs. It helps one understand how functions behave at particular points and makes problem-solving much more manageable.

Simplifying Expressions

Simplifying expressions is a crucial skill when dealing with functions. It involves reducing expressions to their most basic form. When we subtracted the functions \(f(x) = 2x - 5\) and \(g(x) = x + 1\), the expression was:

• \((f-g)(x) = (2x - 5) - (x + 1)\)

To simplify, distribute the negative across each term in \(g(x)\), turning the expression to:

• \((f-g)(x) = 2x - 5 - x - 1\)

Next, combine the like terms, which typically includes combining terms with \(x\) and constant numbers separately. This gives:

• \(x - 6\)

Simplifying helps in revealing the structure of the equations, ensuring calculations are less prone to mistakes. Getting a simplified form makes both evaluating the function with specific values and understanding its graph much easier.