Problem 78
Question
Let \(f(x)=3 x-2\) and \(g(x)=x^{2}+1 .\) Perform each function operation and use the properties of real numbers to justify each step in simplifying your answer. $$ \frac{f(x)}{g(x)} $$
Step-by-Step Solution
Verified Answer
The simplified expression of the division of the functions \(f(x)\) by \(g(x)\) is \(\frac{3x-2}{x^{2}+1}\).
1Step 1: Identify the Functions
Here, the given functions are \(f(x)=3x-2\) and \(g(x)=x^{2}+1\).
2Step 2: Substitute the Values of the Functions
We are required to divide \(f(x)\) by \(g(x)\). Hence, the expression becomes: \(\frac{3x-2}{x^{2}+1}\).
3Step 3: Simplify the Expression
The expression is already in its simplest form and cannot be further simplified.
Key Concepts
Division of FunctionsSimplifying Rational ExpressionsProperties of Real Numbers
Division of Functions
Function division is one way to explore how two functions interact with each other. When given two functions, such as in our example with \(f(x) = 3x - 2\) and \(g(x) = x^2 + 1\), dividing them creates a new function. This technique is writing \(\frac{f(x)}{g(x)}\) which means you are dividing every term of \(f(x)\) by every term of \(g(x)\). It's important to remember that division by zero is undefined. Thus, whatever values make \(g(x)\) equal to zero must be excluded from its domain.
In this expression \(\frac{3x - 2}{x^2 + 1}\), because \(x^2 + 1\) never equals zero for any real number value of \(x\), the division is valid for all real \(x\). This is because the sum of a squared term \((x^2)\) and a positive number \((1)\) is always greater than or equal to 1.
In this expression \(\frac{3x - 2}{x^2 + 1}\), because \(x^2 + 1\) never equals zero for any real number value of \(x\), the division is valid for all real \(x\). This is because the sum of a squared term \((x^2)\) and a positive number \((1)\) is always greater than or equal to 1.
Simplifying Rational Expressions
Simplifying rational expressions is a key skill in algebra that involves reducing complexity while keeping the expression equivalent. In our example \(\frac{3x - 2}{x^2 + 1}\), simplifying means looking to factor both the numerator and the denominator.
In this case, the numerator \(3x - 2\) is already in its simplest linear form and doesn't have common factors with \(x^2 + 1\). The denominator \(x^2 + 1\) is a polynomial that cannot be factored further over the real numbers.
When the numerator and denominator share no common factors, the simplification is complete. It's crucial in mathematical problem-solving to recognize when an expression is already as simple as it can get.
In this case, the numerator \(3x - 2\) is already in its simplest linear form and doesn't have common factors with \(x^2 + 1\). The denominator \(x^2 + 1\) is a polynomial that cannot be factored further over the real numbers.
When the numerator and denominator share no common factors, the simplification is complete. It's crucial in mathematical problem-solving to recognize when an expression is already as simple as it can get.
Properties of Real Numbers
The properties of real numbers are rules that help us understand and manipulate numbers in algebraic expressions. Some fundamental properties include:
In dividing functions as in the exercise \(\frac{f(x)}{g(x)}\), the properties of real numbers assure us that each operation maintains equality and validity across all real values of \(x\) allowed by the domain of the divisor \(g(x)\). Understanding these properties is essential for correctly simplifying and evaluating expressions.
- Commutative Property: The order in which two numbers are added or multiplied does not change the result.
- Associative Property: The way in which numbers are grouped in addition or multiplication does not change their sum or product.
- Distributive Property: Allows you to multiply a sum by multiplying each addend separately and then adding the results.
- Identity Property: An operation with a neutral element, like adding zero.
In dividing functions as in the exercise \(\frac{f(x)}{g(x)}\), the properties of real numbers assure us that each operation maintains equality and validity across all real values of \(x\) allowed by the domain of the divisor \(g(x)\). Understanding these properties is essential for correctly simplifying and evaluating expressions.
Other exercises in this chapter
Problem 78
Let \(f(x)=4 x, g(x)=\frac{1}{2} x+7,\) and \(h(x)=|-2 x+4| .\) Simplify each function. $$ (h \circ(g \circ f))(x) $$
View solution Problem 78
Evaluate each expression. \(_{8} P_{4}\)
View solution Problem 78
Simplify each radical expression. Use absolute value symbols as needed. $$ \sqrt[3]{-64 a^{81}} $$
View solution Problem 78
Expand each binomial. $$ (2 a-b)^{7} $$
View solution