Problem 19 - Precalculus with Limits

Question

In Exercises 17-28, evaluate the indicated function for \(f(x) = x^2 + 1\) and \(g(x) = x - 4\). \((f - g)(0)\)

Step-by-Step Solution

Answer

\((f - g)(0) = 5\

Step 1: Find \(f - g\)

To differentiate between the two functions \(f(x)\) and \(g(x)\), subtract \(g(x)\) from \(f(x)\). Considering \(f(x) = x^2 + 1\) and \(g(x) = x - 4\), the function \(f - g\) is \(f - g = (x^2 + 1) - (x - 4)\)

Step 2: Simplify \(f - g\)

After performing the subtraction of the given functions, you get \(f - g = x^2 + 1 - x + 4 = x^2 - x + 5\)

Step 3: Evaluate \((f - g)(0)\)

Now that the expression for \(f - g\) is found, substitute \(x = 0\) into that expression. Doing so, \((f - g)(0) = (0)^2 - 0 + 5 = 5\)

Key Concepts

Function Subtraction

Function subtraction is the procedure where one algebraic function is subtracted from another. Suppose you are given two functions, such as \(f(x) = x^2 + 1\) and \(g(x) = x - 4\). To find \(f - g\), you need to subtract the entire function \(g(x)\) from \(f(x)\). This is done by subtracting the outputs from each function to create a new function.

• Start by rewriting the function subtraction as \((f - g)(x) = f(x) - g(x)\).
• Substitute the expressions of \(f(x)\) and \(g(x)\) into this formula: \((f - g)(x) = (x^2 + 1) - (x - 4)\).

Avoiding mistakes during function subtraction is vital. Pay special attention to the signs, particularly when subtracting a negative term. Parentheses help to keep track of each part of the subtraction.

Simplifying Expressions

Simplifying expressions is a crucial step after performing operations like function subtraction. Simplification involves combining like terms and making the expression as concise as possible.Once you have \((f - g)(x) = (x^2 + 1) - (x - 4)\), simplify it step-by-step:

• Distribute the minus sign through \(g(x)\). The expression becomes \(x^2 + 1 - x + 4\).
• Combine the like terms by grouping the constant terms and the linear terms together, leading to \(x^2 - x + 5\).

Each part has been simplified, which gets rid of unnecessary complexity. Simplification makes it much easier to handle expressions during further analysis, like evaluating them at specific values.

Evaluating Functions

Evaluating a function means finding the output of a function for a specific input value. This process is useful for determining the exact value of an expression at a particular point.In this exercise, after simplifying \(f - g\) to \(x^2 - x + 5\), you are asked to evaluate it at \(x = 0\):

• Set \(x = 0\) in the expression \(f - g = x^2 - x + 5\).
• Calculate each term individually: \((0)^2 = 0\), \(-0 = 0\), and \(+ 5 = 5\).

Finally, sum these results to find \((f - g)(0) = 5\). This last step confirms the value you sought, showing how algebraic operations and simplification all lead toward understanding a specific numerical outcome.