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Chapter 2: Problem 4

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

Short Answer

Expert verified
(g+f)(3) = 6

Step by step solution

01

- Understand the Functions

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

- Find the Expression for \( (g+f)(x) \)

The expression for \( (g + f)(x) \) is obtained by adding the functions: \[ (g + f)(x) = g(x) + f(x) \] Substitute the given functions: \[ (g + f)(x) = (x^2 - x) + (x - 3) \]
03

- Simplify the Expression

Combine like terms: \[ (g + f)(x) = x^2 - x + x - 3 \] \[ (g + f)(x) = x^2 - 3 \]
04

- Evaluate the Expression at \( x = 3 \)

Substitute \( x = 3 \) into the simplified expression: \[ (g + f)(3) = 3^2 - 3 \] \[ (g + f)(3) = 9 - 3 \]\[ (g + f)(3) = 6 \]

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Key Concepts

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

Function Addition
Function addition is when you combine two functions to create a new function. It's like merging their instructions to see what happens when you follow both. We start with functions \( f(x) \) and \( g(x) \) and want to find \( (g+f)(x) \). We do this by adding the outputs of \( f(x) \) and \( g(x) \). For example, if \( f(x) = x - 3\) and \( g(x) = x^2 - x\), the combined function \( (g + f)(x) \) becomes:

  • First write down \( f(x) \) and \( g(x) \).
  • Add them together: \( (x^2 - x) + (x - 3) \).
  • Combine like terms to simplify: \( x^2 - x + x - 3 \).
  • The simplified expression is: \( x^2 - 3 \).
This new function tells us how the original functions interact when combined.
Simplifying Expressions
Simplifying expressions involves combining and reducing terms to make a math problem easier to understand. Think of it as cleaning up your math work to make it neat.

For instance, consider our function addition result \( (g + f)(x) = x^2 - x + x - 3 \). Here's how we simplify it:
  • Notice that \( -x \) and \( +x \) cancel each other out.
  • This leaves us with \( x^2 - 3 \).
By performing these steps, the expression becomes straightforward and more manageable. Simplifying is crucial for seeing what鈥檚 really happening mathematically.
Evaluating Functions
Evaluating functions means finding the output for a specific input. You plug the value into the function and see what you get.

Given our simplified function from earlier, \( (g + f)(x) = x^2 - 3 \), we want to find \( (g + f)(3) \). Here's how:
  • Substitute \( x \) with 3 in the expression: \( 3^2 - 3 \).
  • Calculate the powers and products: \( 9 - 3 \).
  • Perform the subtraction: \( 6 \).
Therefore, \( (g + f)(3) = 6 \). Evaluating functions is like following a recipe鈥攜ou follow specific steps to get the final result.

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Most popular questions from this chapter

Solve \(|13 x-55|=9\).

Graph each pair of functions (without simplifying the second function) on the same screen of a graphing calculator and explain what each exercise illustrates. a. \(y=x^{4}-x^{2}, y=(-x)^{4}-(-x)^{2}\) b. \(y=x^{3}-x, y=(-x)^{3}-(-x)\) c. \(y=x^{4}-x^{2}, y=(x+1)^{4}-(x+1)^{2}\) d. \(y=x^{3}-x, y=(x-2)^{3}-(x-2)+3\)

Solve each problem. The monthly water bill in Hammond is a function of the number of gallons used. The cost is \(\$ 10.30\) for \(10,000\) gal or less. Over \(10,000\) gal, the cost is \(\$ 10.30\) plus \(\$ 1\) for each 1000 gal over \(10,000\) with any fraction of 1000 gal charged at a fraction of \(\$ 1 .\) On what interval is the cost constant? On what interval is the cost increasing?

Use the minimum and maximum features of a graphing calculator to find the intervals on which each function is increasing or decreasing. Round approximate answers to two decimal places. $$y=-6 x^{2}+2 x-9$$

Find the domain and range of the function \(f(x)=\sqrt{x-2}+3\).
See all solutions

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