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Chapter 2: Problem 3

Let \(f(x)=x-3\) and \(g(x)=x^{2}-x .\) Find and simplify each expression. $$(f+g)(2)$$

Short Answer

Expert verified
(f+g)(2) = 1

Step by step solution

01

Define the Functions

functions are given as: \[ f(x) = x - 3 \] \[ g(x) = x^2 - x\]
02

Add the Functions

Combine the functions by adding them together: \[ (f+g)(x) = f(x) + g(x) \] Simplify: \[ (f+g)(x) = (x - 3) + (x^2 - x) \] \[ (f+g)(x) = x^2 - x + x - 3\] \[ (f+g)(x) = x^2 - 3 \]
03

Evaluate at x = 2

Substitute 2 into the expression \[ (f+g)(2) = (2)^2 - 3 \] \[ (f+g)(2) = 4 - 3 \] \[ (f+g)(2) = 1 \]

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Key Concepts

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

Understanding Function Addition
Function addition involves combining two functions into a new function by simply adding their outputs together. This operation is written as \( (f + g)(x) \). When performing function addition, you will add the corresponding expressions of the given functions.
For example, given the functions \( f(x) = x - 3 \) and \( g(x) = x^{2} - x \), the sum of the functions is:
\[ (f + g)(x) = f(x) + g(x) \] \[ (f + g)(x) = (x - 3) + (x^2 - x) \]
It is important to properly group terms together for simplification.
Here are some key points to remember:
  • Ensure all terms are combined correctly.
  • Pay attention to the signs of each term.
  • Simplify fully by combining like terms.
Evaluation of Functions
Evaluating functions at specific points means substituting a given value of x into the function. This process shows what the function equals when x takes on a particular value.
For example, if we need to evaluate \( (f+g)(2) \), we first add the functions and then plug 2 into the resulting function.
Let's break it down:
We know that \[ (f+g)(x) = x^2 - 3 \] from our previous function addition in Section 1.
Substituting x = 2 gives us:
\[ (f+g)(2) = (2)^2 - 3 \]
Then solve the equation:
\[ (f+g)(2) = 4 - 3 = 1 \]
Key steps include:
  • Identify the function expression.
  • Replace x with the given value.
  • Solve the resulting equation.
Simplification of Expressions
Simplification is the process of making an expression as simple as possible. After performing function operations and evaluations, simplification helps in reducing complex expressions to their most basic form.
In the context of function addition, it involves combining like terms thoroughly.
For instance, when we add \( f(x) = x - 3 \) and \( g(x) = x^2 - x \), we get:
\[ (f+g)(x) = (x - 3) + (x^2 - x) \]
Combining terms:
\[ (f+g)(x) = x^2 - x + x - 3 \]
Notice that \( -x + x = 0 \):
\[ (f+g)(x) = x^2 - 3 \]
Always follow these steps:
  • Group like terms (terms with the same variable and exponent).
  • Combine the coefficients of like terms.
  • Write the simplified expression clearly.
Simplification aids in better understanding and solving further mathematical operations.

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Most popular questions from this chapter

Make a table listing ordered pairs for each function. Then sketch the graph and state the domain and range. Identify any intervals on which \(f\) is increasing, decreasing, or constant. $$f(x)=\sqrt{9-x^{2}}$$

Determine the symmetry of the graph of the function \(f(x)=x^{3}-8 x\).

Determine algebraically whether the function is even, odd, or neither. Discuss the symmetry of each function. $$f(x)=\sqrt{x^{2}+3}$$

Determine algebraically whether the function is even, odd, or neither. Discuss the symmetry of each function. $$f(x)=\left|x^{2}-3\right|$$

Determine algebraically whether the function is even, odd, or neither. Discuss the symmetry of each function. $$f(x)=(x+3)^{2}$$
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