Problem 1 - Algebra and Trigonometry

Question

For the function \(f(x)=\frac{x-1}{x},\) find \(f(2)\) and \(f(3)\)

Step-by-Step Solution

Answer

f(2) = \frac{1}{2}, f(3) = \frac{2}{3}

Step 1 - Understand the Function

The given function is defined as: \[ f(x) = \frac{x-1}{x} \] This function takes an input value for \( x \), subtracts 1 from it, and then divides the result by \( x \).

Step 2 - Substitute x with 2

To find \( f(2) \), substitute \( x \) with 2 in the function: \[ f(2) = \frac{2-1}{2} \]

Step 3 - Simplify the Expression for f(2)

Simplify the expression: \[ f(2) = \frac{1}{2} \] Thus, \( f(2) = \frac{1}{2} \).

Step 4 - Substitute x with 3

To find \( f(3) \), substitute \( x \) with 3 in the function: \[ f(3) = \frac{3-1}{3} \]

Step 5 - Simplify the Expression for f(3)

Simplify the expression: \[ f(3) = \frac{2}{3} \] Thus, \( f(3) = \frac{2}{3} \).

Key Concepts

Function Substitution

When we talk about function substitution, we're referring to the process of replacing the variable in a function with a specific number.
In the given exercise, the function provided is:\[ f(x) = \frac{x-1}{x} \]To evaluate the function for specific values, we simply need to substitute those values for \( x \).
For example, to find \( f(2) \), we replace every \( x \) in the function with 2.This gives us:\[ f(2) = \frac{2-1}{2} \]Similarly, to find \( f(3) \), we replace every \( x \) in the function with 3.This action can be written as:\[ f(3) = \frac{3-1}{3} \]By substituting the values, the function is tailored to specific numbers, making it easier to compute and understand.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form so they're easier to work with.
After substituting the specific values for \( x \) in our function, we get fractions that need to be simplified.Let's look at \( f(2) \):\[ f(2) = \frac{2-1}{2} = \frac{1}{2} \]Here, the subtraction in the numerator gives us 1, and the fraction \( \frac{1}{2} \) is already in its simplest form.
For \( f(3) \):\[ f(3) = \frac{3-1}{3} = \frac{2}{3} \]Similarly, the subtraction in the numerator provides us with 2, and the fraction \( \frac{2}{3} \) is also in its simplest form.
Remember, a fraction is in its simplest form when the numerator and denominator have no common factors other than 1.

Rational Functions

A rational function is a type of function that is defined by the ratio of two polynomials. In our exercise, the function is:\[ f(x) = \frac{x-1}{x} \]Here, the numerator is \( x - 1 \) and the denominator is \( x \).Rational functions can take on a wide range of forms and are often used in various areas of mathematics and applied sciences.
To evaluate rational functions, follow these steps:

• Substitute the given value into the function.
• Simplify the resulting fraction.

In the given examples, substituting \( 2 \) and \( 3 \) for \( x \) provided us with the values \( \frac{1}{2} \) and \( \frac{2}{3} \), respectively. These are simplified results given in fractional form.
Rational functions, therefore, involve both polynomial manipulation and fraction simplification, critical skills in algebra.