Problem 8 - Algebra 2 and Trigonometry

Question

In \(3-8,\) for each function: a. Write an expression for \(f(x) .\) b. Find \(f(5)\) \(x \stackrel{\mathrm{f}}{\rightarrow} \frac{2}{x}\)

Step-by-Step Solution

Answer

a. \( f(x) = \frac{2}{x} \); b. \( f(5) = \frac{2}{5} \).

Step 1: Understanding the Function

The problem describes a function with notation \( x \stackrel{\mathrm{f}}{\rightarrow} \frac{2}{x} \). This means that the function \( f(x) \) is given by \( f(x) = \frac{2}{x} \).

Step 2: Writing the Expression for f(x)

The expression for the function \( f(x) \) is directly given as part of the problem. Therefore, \( f(x) = \frac{2}{x} \). This completes part a of the exercise.

Step 3: Finding f(5)

To find \( f(5) \), substitute \( x = 5 \) into the function expression. This gives:\[ f(5) = \frac{2}{5} \]Therefore, the value of \( f(5) \) is \( \frac{2}{5} \).

Key Concepts

Function Notation

Function notation is a concise way to express the relationship between a dependent variable and one or more independent variables. In the notation \( x \stackrel{\mathrm{f}}{\rightarrow} \frac{2}{x} \), we show that to every value \( x \), the function \( f(x) \) assigns the value \( \frac{2}{x} \). This means that the variable \( x \) is transformed by the function \( f \) into the expression \( \frac{2}{x} \).
Function notation is helpful because it clearly expresses what operation is performed on the input to get the output.

• It tells us the input variable, commonly \( x \).
• It clearly shows the operation: here, division as the main operation.
• It simplifies writing complex expressions and facilitates communication of ideas.

Understanding function notation allows you to evaluate the function at specific points and makes it easier to apply mathematical operations.

Substitution

Substitution is the process of replacing a variable within an expression or function with a given value. In our problem, this allows us to find specific outputs of the function by replacing \( x \) with any particular number, like 5. For example, to find \( f(5) \) for the function \( f(x) = \frac{2}{x} \), you would substitute \( x \) with 5 to get \( \frac{2}{5} \).
Substitution plays a key role in:

• Sorting out specific values of a function's output.
• Testing hypotheses about a function or proving certain properties.
• Simplifying expressions by replacing variables with numerical values.

By executing substitution steps accurately, you can solve for unknowns or find the function's output for any input value efficiently.

Rational Expressions

A rational expression is a fraction that has polynomials in both its numerator and its denominator. In the expression \( \frac{2}{x} \), 2 is a constant polynomial and \( x \) is a linear polynomial. Rational expressions can be simplified, evaluated, or manipulated in similar ways to fractions.
Key points about rational expressions include:

• They can be undefined when the denominator equals zero. For \( \frac{2}{x} \), \( x eq 0 \).
• Simplification can involve cancelling out terms, factoring, or finding common denominators.
• Applications include simplifying expressions, solving equations, and modeling real-world scenarios.

Understanding rational expressions is important, as they appear frequently in algebra, calculus, and in various applied mathematical fields.