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Chapter 3: Problem 41

Using the functions \(f(x)=2 x+3\) and \(g(x)=x^{2}-1,\) find the following. $$(f+g)(-3)$$

Short Answer

Expert verified
\( (f+g)(-3) = 5 \)

Step by step solution

01

Add the two functions f(x) and g(x)

To find the sum (f+g)(x), we will add the functions f(x) and g(x) together: \[ (f+g)(x) = f(x) + g(x) \] Using the given functions, plug in their respective equations: \[(f+g)(x) = (2x + 3) + (x^2 - 1)\]
02

Simplify the sum of the functions

Combine the like terms and simplify the expression for (f+g)(x): \[(f+g)(x) = x^2 + 2x + 3 - 1\] \[(f+g)(x) = x^2 + 2x + 2\]
03

Evaluate the sum of the functions at x = -3

Now that we have the simplified expression for (f+g)(x), we can substitute x = -3 to find the final answer: \[(f+g)(-3) = (-3)^2 + 2(-3) + 2\]
04

Calculate the final answer

Perform the arithmetic operations within the expression: \[(f+g)(-3) = 9 - 6 + 2\] \[(f+g)(-3) = 3 + 2\] \[(f+g)(-3) = 5\] So, \((f+g)(-3) = 5\).

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Key Concepts

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

Addition of Functions
When you're faced with two functions, such as \( f(x) = 2x + 3 \) and \( g(x) = x^2 - 1 \), often you'll want to combine them into a single function. This is accomplished through function addition. By adding two functions, you essentially add their outputs for the same input value.

Here's how you do it:
  • First, write down the expressions for both functions.
  • Then, simply add the corresponding expressions together.
For our example, this looks like:
  • First, identify each function: \( f(x) = 2x + 3 \) and \( g(x) = x^2 - 1\).
  • Add them to form a new function: \( (f+g)(x) = (2x + 3) + (x^2 - 1) \).
This results in a new function, \( (f+g)(x) = x^2 + 2x + 2 \), whose form can now be used for further calculations.
Evaluating Functions
Once you have a function like \( (f+g)(x) = x^2 + 2x + 2 \), the next step is often to find the function's value at a specific input, or to evaluate the function.

Evaluation simply means plugging a given value into the function in place of \( x \). This calculation gives you the output or the function's value for that particular input.
  • Substitute the chosen value of \( x \) into the function.
  • Calculate step-by-step to simplify the arithmetic operations.
For \( (f+g)(-3) \):
  • Substitute \( x = -3 \) into the function: \( (-3)^2 + 2(-3) + 2 \).
  • Calculate: \( 9 - 6 + 2 \).
  • Simplify: \( 3 + 2 \).
  • Finally, find: \( (f+g)(-3) = 5 \).
This tells us that when \( x = -3 \), the function \( (f+g)(x) \) results in the value 5.
Polynomial Simplification
In mathematics, combining like terms is something you'll encounter frequently鈥攖his process is part of simplifying polynomials. When you create a new function from two functions, like in our case with \( (f+g)(x) \), polynomial simplification is necessary.

Simplifying involves combining similar terms to make the expression cleaner and easier to work with.

Here's how you go about it:
  • Look at each term in the function's sum: \( x^2 + 2x + 3 - 1 \).
  • Combine constants: \( +3 - 1 = 2 \).
  • Rewrite the function in its simplest form: \( x^2 + 2x + 2 \).
The result is a simplified polynomial, which makes it more straightforward when you're evaluating the function later. Keeping expressions simplified reduces errors and makes calculations faster, especially when multiple steps or complex expressions are involved.

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Most popular questions from this chapter

(Easter Sunday) The date for Easter Sunday in any year \(y\) can be computed as follows. Let \(a=y \bmod 19, b=y \bmod 4, c=y \bmod 7, d=(19 a+24)\) \(\bmod 30, e=(2 b+4 c+6 d+5) \bmod 7,\) and \(r=(22+d+e) .\) If \(r \leq 31,\) then Easter Sunday is March \(r ;\) otherwise, it is April \([r(\bmod 31)] .\) Compute the date for Easter Sunday in each year. $$2000$$

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Just as \(\sum\) is used to denote sums, the product \(a_{k} a_{k+1} \ldots a_{m}\) is denoted by \(\prod_{i=k}^{m} \mathrm{a}_{i} .\) The product symbol \(\Pi\) is the Greek capital letter \(p i .\) For example, \(n !=\prod_{i=1}^{n} i .\) Evaluate each product. $$\prod_{j=-5}^{50} 1$$

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