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Chapter 1: Problem 13

Evaluate the indicated function, where \(f(x)=x^{2}-3 x+2\) and \(g(x)=2 x-4\). $$(f+g)(5)$$

Short Answer

Expert verified
The value of (f+g)(5) is 18.

Step by step solution

01

Write down the given functions

First, let's write down the two functions as given in the problem: for \(f(x)\) = \(x^{2}-3x+2\) and for \(g(x)\) = \(2x-4\). We are to find the sum of these two functions, \(f+g\). This typically means adding the two function like terms.
02

Add the two functions

(f+g)(x) = \(f(x) + g(x)\). This translates to \(f(x) + g(x) = (x^{2}-3x+2) + (2x-4)\). You will notice that this sums up to \(f(x) + g(x) = x^{2}-x-2\).
03

Evaluate the sum function at x = 5

Now substitute \(x=5\) into the sum function obtained in Step 2. This infrastucture is \(f(5) + g(5) = (5^{2}-5-2) = 18\). So, (f+g)(5) equals to 18.

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Key Concepts

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

Polynomial Functions
Polynomial functions are fundamental components of algebra and encompass expressions made up of variables, coefficients, and non-negative integer exponents. Each term within a polynomial function is made up of a coefficient and a variable, like in the expression \(x^2 - 3x + 2\). This specific function is a quadratic polynomial because the highest exponent is 2.
  • The general form of a polynomial is \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where \(a_n\) is not equal to zero.
  • The degree of the polynomial is the highest power of the variable, which decides the curve's shape when graphed.
  • Polynomial functions can exhibit diverse graph shapes, including parabolas, which are characteristic of quadratic functions.
Understanding polynomial functions is crucial as they form the foundation for modeling and solving real-world problems in mathematics and science.
Function Addition
Function addition is an operation where you combine two functions by adding their corresponding terms. If you have two functions, such as \(f(x) = x^2 - 3x + 2\) and \(g(x) = 2x - 4\), adding these functions together results in a new, combined function.
  • To perform function addition, simply add the coefficients of like terms (terms with the same powers).
  • For our example, adding \(f(x)\) and \(g(x)\) yields: \((x^2 - 3x + 2) + (2x - 4) = x^2 - x - 2\).
  • This new function, \(x^2 - x - 2\), represents the sum of the two functions.
Function addition is a straightforward yet powerful tool in algebra, allowing the combination of different linear and non-linear relationships.
Substitution in Functions
Substitution in functions involves replacing the variable in a function with a specific value to evaluate the function. This is an essential technique for finding the value of a function at a particular point. For example, substituting \(x = 5\) into the function \(f(x) + g(x)\).
  • Start by substituting \(x = 5\) into the combined function: \((f+g)(x) = x^2 - x - 2\).
  • Proceed to solve: \((5)^2 - 5 - 2 = 25 - 5 - 2\).
  • The result of the calculation is 18, meaning \((f+g)(5) = 18\).
Substitution in functions helps match mathematical models to real-life situations by determining exact values, and is crucial in fields ranging from physics to economics.

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Most popular questions from this chapter

Solve \(3 x-5 y=15\) for \(y\)

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