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Chapter 9: Problem 21

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1\) \(f(-x)\)

Short Answer

Expert verified
f(-x) = 3x + 4

Step by step solution

01

- Understand the Function

Given the function definitions: \(f(x) = -3x + 4\) and \(g(x) = -x^2 + 4x + 1\). We need the function value for \(f(-x)\).
02

- Substitute \(x\) with \(-x\) in f(x)

Substitute \(x\) with \(-x\) in the function \(f(x)\) to find \(f(-x)\): \(f(-x) = -3(-x) + 4\).
03

- Simplify the Expression

Simplify the expression obtained after substitution: \(-3(-x) + 4 = 3x + 4\). Thus, \(f(-x) = 3x + 4\).

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Key Concepts

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

Substitution in Functions
Substitution in functions is a crucial technique used in various areas of mathematics. It involves replacing a variable within a function with a different expression. This method helps determine how changes in the input variable affect the output. For example, if we have a function defined as \(f(x) = -3x + 4\) and we want to find the value of \(f(-x)\), we substitute \(x\) with \(-x\).

In our case:
  • Original function: \(f(x) = -3x + 4\)
  • Substituting \(x\) with \(-x\): \(f(-x) = -3(-x) + 4\)
  • Simplifying gives: \(f(-x) = 3x + 4\)
This example shows how substitution helps in understanding the behavior of a function when the sign or value of the variable changes.
Simplifying Algebraic Expressions
Simplifying algebraic expressions is the process of making a complicated expression more manageable and easier to understand. This typically involves combining like terms and performing basic arithmetic operations. In our example, we simplified the expression obtained after substitution.

After substituting \(x\) with \(-x\):
  • We had: \(f(-x) = -3(-x) + 4\)
  • Simplify \(-3(-x)\): When you multiply a negative number by another negative number, the result is positive. So, \(-3(-x) = 3x\).
  • Our simplified expression is: \(f(-x) = 3x + 4\).
Understanding each step in the simplification process ensures you can handle more complex algebraic manipulations in the future.
Polynomial Functions
Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients.

A general polynomial in one variable \(x\) can be written as:
  • \(P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\)
  • Where \(a_n, a_{n-1}, \, ... \, , a_1, a_0\) are constants, and \(n\) is a non-negative integer.
In our exercise, we saw two polynomial functions:
  • \(f(x) = -3x + 4\)
  • \(g(x) = -x^2 + 4x + 1\)
The function \(f(x)\) is a polynomial of degree 1 because the highest power of \(x\) is 1. On the other hand, \(g(x)\) is a polynomial of degree 2 because the highest power of \(x\) is 2. When working with polynomial functions, knowing how to identify the degree and how to simplify expressions is fundamental for solving more complex mathematical problems.

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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?

A pair of shoes is marked \(50 \%\) off. A customer has a coupon for an additional \(\$ 10\) off. (a) Write a function \(g\) that finds \(50 \%\) of \(x\). (b) Write a function \(f\) that subtracts 10 from \(x\). (c) Write and simplify the function \((f \circ g)(x)\). (d) Use the function from part (c) to find the sale price of a pair of shoes that has an original price of \(\$ 100\).

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following $$ (f g)(0) $$

Determine whether each relation defines y as a function of \(x .\) (Solve for y first if necessary.) Give the domain. $$ y=x $$

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