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Chapter 5: Problem 20

Let \(f(x)=-3 x+1\) and \(g(x)=x^{2}+2 .\) Find each of the following. $$ (g / f)(x) $$

Short Answer

Expert verified
\( \frac{x^2 + 2 }{-3x + 1} \)

Step by step solution

01

- Understand the Function Operation

The problem asks us to find \( \frac{g}{f}(x) \). This means that we need to divide the function \( g(x) \) by the function \( f(x) \).
02

- Substitute the Functions

Start by substituting the given functions into the expression: \( \frac{g(x)}{f(x)} \). We know that \( g(x) = x^2 + 2 \) and \( f(x) = -3x + 1 \).
03

- Write the Expression

Now our expression looks like this: \( \frac{x^2 + 2}{-3x + 1} \).
04

- Simplify if Possible

Check to see if we can simplify the fraction further by canceling any common factors. Here, no further simplification is possible because \( x^2 + 2 \) and \( -3x + 1 \) do not share common factors.

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Key Concepts

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

function division
Function division involves dividing one function by another. To do this, you place one function's output over the other function's output as a fraction. Here, we are asked to find \(\frac{g}{f}(x)\), which means dividing \(g(x)\) by \(f(x)\). This operation allows us to understand how one function behaves relative to another when both have the same input value.
Start by substituting the specific functions into the expression. Given \(g(x) = x^2 + 2 \) and \(f(x) = -3x + 1\), our divided function looks like this: \( \frac{x^2 + 2}{-3x + 1} \).
Make sure to understand that this forms a new function based on the operations defined by both original functions.
algebraic expressions
Algebraic expressions involve combining constants, variables, and arithmetic operations. In our example, these expressions are \(x^2 + 2\) and \(-3x + 1\).
When working with algebraic expressions, always pay attention to:
  • Coefficients: These are the numerical factors in terms like \(-3\) in \(-3x\).
  • Variables: These are the symbols representing numbers, like \x\ in both of our expressions.
  • Constants: These are fixed values, such as \2\ and \1\ in our case.
  • The structure of expressions: Ensure you're clear about the addition, subtraction, multiplication, and division occurring within the expression.
Understanding these basics is crucial to manipulating and simplifying these expressions when necessary.
simplifying fractions
Simplifying fractions in algebra involves reducing the fraction to its simplest form if possible. This means cancelling out any common factors from the numerator and the denominator.
In our function \( \frac{x^2 + 2}{-3x + 1} \), we first check for common factors in the numerator and the denominator. Since \(x^2 + 2\) and \(-3x + 1\) do not have any common factors, the expression remains in its simplest form.
Here are some tips to remember when simplifying:
  • Look for common factors between the numerator and the denominator.
  • You can only cancel factors, not terms connected by addition or subtraction.
  • If there are no common factors, your fraction is already in its simplest form.
Simplifying fractions helps in making expressions easier to understand and solve.

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Most popular questions from this chapter

Replace \(\square\) with \(>,<,\) or \(=\) to write a true sentence. $$ 4^{3} \square 5^{3} $$

Find the domain of \(F / G,\) if $$ F(x)=\frac{1}{x-4} \quad \text { and } \quad G(x)=\frac{x^{2}-4}{x-3} $$

A grain of sand is placed on the first square of a chessboard, two grains on the second square, four grains on the third, eight on the fourth, and so on. Without a calculator, use scientific notation to approximate the number of grains of sand required for the 64 th square. (Hint: Use the fact that \(\left.2^{10} \approx 10^{3} \cdot\right)\)

Write the reciprocal of \(8.00 \times 10^{-23}\) in scientific notation.

Write \(\frac{4}{32}\) in decimal notation, simplified fraction notation, and scientific notation.
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