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Chapter 9: Problem 40

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following. $$ (f-g)(x) $$

Short Answer

Expert verified
(f - g)(x) = x^2 - 2x - 9

Step by step solution

01

Identify the functions

Given functions are:\(f(x) = x^2 - 9\)\(g(x) = 2x\)\(h(x) = x - 3\)
02

Define the operation

We need to find \((f - g)(x)\), which represents the subtraction of function \(g(x)\) from function \(f(x)\).
03

Subtract the functions

To find \((f - g)(x)\), subtract \(g(x)\) from \(f(x)\):\(\begin{aligned} (f - g)(x) &= f(x) - g(x) \ &= (x^2 - 9) - 2x \end{aligned}\)
04

Combine and simplify

Simplify the expression:\((f - g)(x) = x^2 - 2x - 9\)

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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 mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients.
They are written in the general form: \[ P(x) = a_0 + a_1x + a_2x^2 + ... + a_nx^n \]
Here, each \( a_i \) is a coefficient, and \( x^i \) is the variable raised to a power \( i \) (which is a non-negative integer).
For instance, consider the function given in the exercise, \( f(x) = x^2 - 9 \).
This is a polynomial of degree 2 because the highest power of \( x \) is 2. You can identify polynomials by looking for terms with variables raised to non-negative integer powers.
They could be simpler like \( g(x) = 2x \), which is a polynomial of degree 1.
Algebraic Operations
Algebraic operations include addition, subtraction, multiplication, and division of algebraic expressions.
These operations help simplify expressions and solve equations. In the context of functions, these operations are applied to the expressions that define the functions.
In the exercise, we focus on subtraction of two functions, \((f - g)(x)\). Here's a step-by-step way to understand it:
  • Start by writing down the functions: \(f(x) = x^2 - 9\) and \(g(x) = 2x\).
  • Identify the operation required: \(f - g\). This means we need to subtract \( g(x) \) from \( f(x) \).
  • Combine the functions by subtracting their expressions: \((x^2 - 9) - 2x\).
  • Simplify the resulting expression: \(x^2 - 2x - 9\).
This demonstrates the power of algebraic operations in manipulating and simplifying expressions.
Function Notation
Function notation is a way to represent functions in the form of \( f(x) \).
This simplifies the process of working with functions and indicates how the input \( x \) is transformed into an output.
For instance, in our exercise, \( f(x) \) represents \( x^2 - 9 \), which means for any value of \( x \), the function returns \( x^2 - 9 \).
Here鈥檚 how to understand the notation and operations on functions:
  • Identify each function using its notation, e.g., \( f(x) = x^2 - 9 \) and \( g(x) = 2x \).
  • When you see \( (f - g)(x) \), it means you鈥檙e applying the operation (subtraction in this case) to the functions.
  • Using function notation, write the operation and simplify: \( f(x) - g(x) = (x^2 - 9) - 2x \).
  • This simplifies to \( x^2 - 2x - 9 \).
Function notation makes it easier to understand and visualize these operations in a structured way.

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Most popular questions from this chapter

Solve each problem. The amount of water emptied by a pipe varies directly as the square of the diameter of the pipe. For a certain constant water flow, a pipe emptying into a canal will allow 200 gal of water to escape in an hour. The diameter of the pipe is 6 in. How much water would a 12 -in. pipe empty into the canal in an hour, assuming the same water flow?

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following. $$ (g+h)\left(\frac{1}{4}\right) $$

The cost in dollars to produce \(x\) youth baseball caps is \(C(x)=4.3 x+75 .\) The revenue in dollars from sales of \(x\) caps is \(R(x)=25 x\) (a) Write and simplify a function \(P\) that gives profit in terms of \(x\). (b) Find the profit if 50 caps are produced and sold.

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following. $$ \left(\frac{h}{g}\right)(x) $$

For each pair of functions, find \(\left(\frac{f}{g}\right)(x)\) and give any \(x\) -values that are not in the domain of the quotient function. $$ f(x)=4 x^{2}-23 x-35, \quad g(x)=x-7 $$
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