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Chapter 2: Problem 86

Given \(f(x)=x-3\) and \(g(x)=x^{2}+3 x+9,\) find \((f g)(x)\) [1.7]

Short Answer

Expert verified
\((f \circ g)(x) = x^2 + 3x + 6\)

Step by step solution

01

Understand the Composition of Functions

The composition of two functions \(f(x)\) and \(g(x)\) is denoted as \((f \circ g)(x)\), which is equivalent to \(f(g(x))\). This means that you substitute the function \(g(x)\) into the function \(f(x)\).
02

Substitute g(x) into f(x)

The function \(g(x)\) is given as \(x^2 + 3x + 9\). Therefore, when \(g(x)\) is substituted into \(f(x)\), it results in \(f(g(x)) = (x^2 + 3x + 9) - 3\)
03

Simplify the Result

After substituting and performing the subtraction, the expression simplifies to \(x^2 + 3x + 6\).

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Key Concepts

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

Precalculus
Precalculus serves as the foundation for understanding more complex mathematical topics. It's essential because it introduces concepts such as functions, algebra, and geometry, preparing students for calculus. In this context, functions become particularly important as they model relationships between variables. During precalculus, students explore function composition.
  • Function composition involves combining functions to create a new function. Substituting one function into another enables examining how changes in one variable affect another.
  • This exercise exemplifies composition by inputting one polynomial function into another.
Comprehending these fundamentals is crucial as they underpin further mathematical studies. Thorough grasp of precalculus enhances problem-solving skills and contributes to success in future math courses.
Polynomials
Polynomials are algebraic expressions made up of terms consisting of variables and coefficients. These terms can involve addition, subtraction, and non-negative integer exponents. Understanding polynomials is important in both algebra and calculus, as they form the basis for many other algebraic concepts.
  • The given function in the exercise, \(g(x) = x^2 + 3x + 9\), is a polynomial of degree 2. It includes a quadratic term, a linear term, and a constant term.
  • In this exercise, function composition involves substituting \(g(x)\) into \(f(x)\), showing how polynomial expressions can be combined to form a new expression.
Polynomials serve as a critical tool in various fields such as physics, economics, and engineering, making them vital components of a mathematical curriculum.
Algebraic Functions
Algebraic functions consist of polynomials and various algebraic operations, such as addition, subtraction, multiplication, division, and composition, among others. Grasping algebraic functions helps in understanding how different expressions are connected and how they influence each other.
  • This exercise showcases the algebraic operation of composition. The function \(f(x) = x - 3\) is applied to the polynomial \(g(x)\).
  • Interpreting the new expression \(f(g(x)) = x^2 + 3x + 6\) involves recognizing how the function's operations affect the variables' relationships.
Mastering algebraic functions is invaluable, equipping students with tools to solve complex problems and develop deeper insights into a wide array of mathematical scenarios.

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Most popular questions from this chapter

Find a polynomial function of lowest degree with integer coefficients that has the given zeros. $$3+i, 3-i, 2+5 i, 2-5 i$$

In Exercises 61 to 70 , use the quadratic formula to solve each quadratic equation. $$x^{2}-4 x=1$$

In Exercises 61 to 70 , use the quadratic formula to solve each quadratic equation. $$2 x^{2}+2 x+13=0$$

In Exercises 51 to 60 , take square roots to solve each quadratic equation. $$(x+2)^{2}+5=0$$

Determine the degree of the numerator and the degree of the denominator of \(\frac{x^{3}+3 x^{2}-5}{x^{2}-4} \cdot[\mathrm{A} .2]\)
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