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Chapter 2: Problem 1

Let \(f(x)=x^{2}+3\) and \(g(x)=-2 x+6 .\) Find each of the following. $$(f+g)(3)$$

Short Answer

Expert verified
12

Step by step solution

01

- Evaluate f(x) at x=3

First, substitute 3 into the function \(f(x)\). This gives: \[ f(3) = 3^2 + 3 \] Calculate the result: \[ f(3) = 9 + 3 = 12 \]
02

- Evaluate g(x) at x=3

Next, substitute 3 into the function \(g(x)\). This gives: \[ g(3) = -2(3) + 6 \] Calculate the result: \[ g(3) = -6 + 6 = 0 \]
03

- Add the results of f(3) and g(3)

Now, add the values obtained from \(f(3)\) and \(g(3)\): \[ (f + g)(3) = f(3) + g(3) \] Substitute the results found earlier: \[ (f + g)(3) = 12 + 0 = 12 \]

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Key Concepts

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

Composite Functions
A composite function is created when one function is applied to the result of another function. If we have two functions, say \(f(x)\) and \(g(x)\), the composite function is denoted as \( (f \circ g)(x) = f(g(x)) \). This notation means we first apply \(g(x)\), then use the output as the input for \(f(x)\).
For example, if you have \(f(x) = x^2 + 3\) and \(g(x) = -2x + 6\), you can form a composite function \( (f \circ g)(x) \). Here's how it works:
  • First, apply \(g(x)\): \( g(x) = -2x + 6 \).
  • Then, apply \(f\) to the result of \(g(x)\): \( f(g(x)) = f(-2x + 6) \).
Plug \( g(x) = -2x + 6 \) into \(f\):
\[ f(-2x + 6) = (-2x + 6)^2 + 3 \].
Simplify the expression to find the composite function's formula.
Adding Functions
Adding functions is straightforward. It involves combining two functions by adding their respective outputs. If \(f(x)\) and \(g(x)\) are two functions, their sum is written as \( (f + g)(x) \). This means \(\text{for any x}\), \((f + g)(x) = f(x) + g(x)\).
Let's consider \(f(x) = x^2 + 3\) and \(g(x) = -2x + 6\).
To find \((f + g)(3)\):
  • Evaluate \(f(x)\) at \(x=3\): \[ f(3) = 3^2 + 3 = 12 \]
  • Evaluate \(g(x)\) at \(x=3\): \[ g(3) = -2(3) + 6 = 0 \]
  • Add the results: \[ (f + g)(3) = f(3) + g(3) = 12 + 0 = 12 \]
This is how you add functions at specific points. The key is to evaluate each function separately and then combine the results.
Evaluating Functions
Evaluating functions means finding the function's value for a specific input. For a function \(f(x)\), evaluating it at \(x=3\) means calculating \(f(3)\). This involves substituting \(x\) with the number 3 in the function's formula.
Consider these steps using \(f(x) = x^2 + 3\) and \(g(x) = -2x + 6\):
For \(f(x)\) at \(x=3\): Substituting \(3\) into \(f(x)\) we get:
  • \[ f(3) = 3^2 + 3 = 9 + 3 = 12 \]
For \(g(x)\) at \(x=3\): Substitute \(3\) into \(g(x)\):
  • \[ g(3) = -2(3) + 6 = -6 + 6 = 0 \]
These are the function values at that specific input. Remember, when evaluating, always substitute and then simplify the result.

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Most popular questions from this chapter

Let \(f(x)=x^{2}+3\) and \(g(x)=-2 x+6 .\) Find each of the following. $$\left(\frac{f}{g}\right)(5)$$

Given functions \(f\) and \(g,\) find ( \(a\) ) \((f \circ g)(x)\) and its domain, and ( \(b\) ) \((g \circ f)(x)\) and its domain. See Examples 6 and 7 . $$f(x)=\sqrt{x}, \quad g(x)=x+3$$

Find all values of \(y\) such that the distance between \((3, y)\) and \((-2,9)\) is 12.

Let \(f(x)=2 x-3\) and \(g(x)=-x+3 .\) Find each function value. $$(g \circ f)(0)$$

Write an equation ( \(a\) ) in standard form and ( \(b\) ) in slope-intercept form for the line described. See Example \(6 .\) Find \(k\) so that the line through \((4,-1)\) and \((k, 2)\) is (a) parallel to \(3 y+2 x=6\) (b) perpendicular to \(2 y-5 x=1\)
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