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Chapter 5: Problem 88

Given \(f(x)=2 x-3\) and \(g(x)=x 2+3 x-1,\) find the following. $$ (g-f)(12) $$

Short Answer

Expert verified
(g-f)(12) = 158

Step by step solution

01

Understand the Operation

\((g - f)(x)\) means we have to subtract the function \(f(x)\) from \(g(x)\). This operation results in a new function described as \((g-f)(x) = g(x) - f(x)\).
02

Substitute the Functions

Substitute the given functions into \((g-f)(x)\):\[ (g-f)(x) = g(x) - f(x) = (x^2 + 3x - 1) - (2x - 3) \]
03

Simplify the Expression

To simplify \((x^2 + 3x - 1) - (2x - 3)\), distribute the minus sign and combine like terms:\[(g-f)(x) = x^2 + 3x - 1 - 2x + 3 = x^2 + x + 2\]
04

Evaluate at x = 12

Substitute \(x = 12\) into the function:\[(g-f)(12) = 12^2 + 12 + 2\]
05

Calculate the Result

Calculate each term:- \(12^2 = 144\)- \(12 = 12\)- \(2 = 2\)Sum:\[ (g-f)(12) = 144 + 12 + 2 = 158 \]

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Key Concepts

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

Function evaluation
Function evaluation is a simple way to find the output value of a function by substituting a specific input value into the function. When you have a function like \(f(x)\), replacing \(x\) with a number provides a concrete result.
  • For example, with the function \(f(x) = 2x - 3\), evaluating at \(x = 12\) involves replacing \(x\) with 12.
  • Plug in the value: \(f(12) = 2 \times 12 - 3\).
  • Perform the calculation: \(24 - 3 = 21\).

This process helps you understand what a function outputs for specific inputs, which is a key first step in algebra.
Polynomial functions
Polynomial functions are expressions involving variables raised to whole number powers. A polynomial might look complex, but it is composed of simple terms added or subtracted.
  • A typical polynomial function looks like \(g(x) = x^2 + 3x - 1\).
  • It consists of terms: a quadratic term \(x^2\), a linear term \(3x\), and a constant \(-1\).
  • Each term has a coefficient (the number in front of the variable) and a degree (the exponent of the variable).

Understanding polynomials is essential for working with complex algebraic expressions.
Algebraic expressions
Algebraic expressions are combinations of numbers, variables, and operations like addition, subtraction, multiplication, and division.
  • An example is \((g - f)(x) = x^2 + 3x - 1 - (2x - 3)\).
  • To simplify, distribute any minus signs across terms and combine like terms.
  • In our example: \(x^2 + 3x - 1 - 2x + 3\) simplifies to \(x^2 + x + 2\).

Mastering these expressions allows you to simplify and solve equations, forming the backbone of algebra.

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