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Chapter 8: Problem 68

Given \(f(x)=6 x+5\) and \(g(x)=x^{2}-3 x+2,\) find each of the following: a. \((f \circ g)(x)\) b. \((g \cdot f)(x)\) c. \((f \circ g)(-1)\)

Short Answer

Expert verified
The solutions are: a. The composed function \( (f \circ g)(x) = 6x^{2}-18x+17 \), b. The multiplied function \( (g \cdot f)(x) = 6x^{3}-13x^{2}-3x+10 \), and c. The value of the composed function at -1 is \( (f \circ g)(-1) = 41 \).

Step by step solution

01

Find (f 鈭� g)(x)

Function composition \( (f \circ g)(x) \) means we plug g(x) into f(x). That means we substitute \( x^{2}-3 x+2 \) into \( 6x+5 \). That gives us \( f(g(x)) = 6(g(x)) + 5 = 6(x^{2}-3 x+2) + 5 = 6x^{2}-18x+12 + 5 = 6x^{2}-18x+17 \).
02

Find (g 鈰� f)(x)

Function multiplication \( (g \cdot f)(x) \) means we multiply g(x) by f(x). So, we get \( g(x) \cdot f(x) = (x^{2}-3x+2) \cdot (6x+5) = 6x^{3} -18x^{2} + 12x +5x^{2} -15x +10 = 6x^{3}-13x^{2}-3x+10 \).
03

Find (f 鈭� g)(-1)

We have already calculated \( (f \circ g)(x) \) in step 1, so now plug in x = -1 into the derived function. Substitute -1 into \( 6x^{2}-18x+17 \) to get \( (f \circ g)(-1) = 6(-1)^{2}-18(-1)+17 = 6+18+17 = 41 \).

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Key Concepts

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

Function Composition
Function composition is all about combining two functions. When you see
  • \((f \circ g)(x)\)
it means you need to put one function inside another. Essentially, you take the output of one function and use it as the input for the next function!
This is like following a two-step process. Let's break it down with our given functions:- **Defined Functions**:
  • \(f(x) = 6x + 5\)
  • \(g(x) = x^2 - 3x + 2\)
- **Perform the Composition**:
  • First, you compute \(g(x)\), which is \(x^2 -3x + 2\).
  • Next, you replace every \(x\) in \(f(x)\) with \(g(x)\). So, \(f(g(x)) = 6(g(x)) + 5\).
  • This turns into \(6(x^2 - 3x + 2) + 5\).
      • Once you simplify, you'll get the result: \(6x^2 - 18x + 17\).

      This shows that the composition of functions gives you a new polynomial function!
Polynomial Functions
Polynomial functions are mathematical expressions consisting of variables raised to positive integer powers and their coefficients. Here's a brief breakdown to help you understand them:- **Basics of Polynomial Functions**:
  • A typical polynomial can look like: \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where \(a_n, a_{n-1}, \ldots, a_0\) are constants.
  • Each piece, such as \(a_nx^n\), is called a term.
- **Our Example**:
  • \(g(x) = x^2 - 3x + 2\) is a polynomial function of degree 2.
  • \(f(x) = 6x + 5\) is a polynomial function of degree 1.
- **Combining Polynomials**:
  • When functions are combined, as in \((g \cdot f)(x)\), you multiply the polynomials together.
  • The result, \(6x^3 - 13x^2 - 3x + 10\), is another polynomial function, now of degree 3.
Polynomial functions are incredibly versatile and serve as building blocks for many more complex functions you'll encounter.
Substitution Method
Substitution is a straightforward but crucial method in mathematics used to simplify expressions or solve equations. When working with function operations, substitution comes handy in two primary ways:- **Within Functions**:
  • To substitute, simply replace the variable (like \(x\)) in a function by another expression or a specific number.
  • In the solution, we used substitution to find \((f \circ g)(-1)\).
  • This meant taking the result of \((f \circ g)(x)\) and plugging \(x = -1\) into it.
      • - **Our Calculation**:
      • Start with the derived function \(6x^2 - 18x + 17\).
      • Substitute \(-1\) for \(x\): \(6(-1)^2 - 18(-1) + 17\).
      • By solving this, you find the value: \(41\).


      Substitution is an essential method to know, allowing you to evaluate the values of composed functions and approach more complex problems confidently.

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Most popular questions from this chapter

Suppose that \(\sin \alpha=\frac{3}{5}\) and \(\cos \beta=-\frac{12}{13}\) for quadrant II angles \(\alpha\) and \(\beta .\) Find the exact value of each of the following: a. \(\cos \alpha\) b. \(\sin \beta\) c. \(\cos (\alpha+\beta)\) d. \(\sin (\alpha+\beta)\) (Section \(6.2, \text { Example } 5)\)

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function Constraints $$ \begin{aligned} &z=2 x+3 y\\\ &\left\\{\begin{array}{l} {x \geq 0, y \geq 0} \\ {2 x+y \leq 8} \\ {2 x+3 y \leq 12} \end{array}\right. \end{aligned} $$

The function $$f(t)=\frac{25.1}{1+2.7 e^{-0.05 t}}$$ models the population of Florida, \(f(t),\) in millions, \(t\) years after \(1970 .\) a. What was Florida's population in \(1970 ?\) b. According to this logistic growth model, what was Florida's population, to the nearest tenth of a million. in \(2010 ?\) Does this underestimate or overestimate the actual 2010 population of 18.8 million? c. What is the limiting size of the population of Florida?

Write a system of equations having {(-2, 7)} as a solution set. (More than one system is possible.)

Use the exponential growth model, \(A=A_{0} e^{k t},\) to solve this exercise. In \(1975,\) the population of Europe was 679 million. By \(2015,\) the population had grown to 746 million. a. Find an exponential growth function that models the data for 1975 through 2015 b. By which year, to the nearest year, will the European population reach 800 million? (Section \(4.5,\) Example 1 )
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