Question

۱۰) براي تابع خطي (f) داريم (f(2) = -1) و (f(3) = 2) است، نمايش جبري تابع (f) را بدست آوريد؟

Explanation:

🆕 New Concept Discovered: Linear Functions
Finding the equation of a straight line

Step 1: Identify the general form of a linear function

A linear function \( f(x) \) can always be written in the slope-intercept form:

$$ f(x) = mx + b $$

where \( m \) represents the slope (rate of change) and \( b \) represents the y-intercept.

We are given two points on this line:

1. \( f(2) = -1 \implies (2, -1) \)
2. \( f(3) = 2 \implies (3, 2) \)

Step 2: Calculate the slope (\( m \))

The slope \( m \) measures how steep the line is. It is the change in the output values divided by the change in the input values:

$$ m = \frac{f(3) - f(2)}{3 - 2} $$

Substitute the given values:

$$ m = \frac{2 - (-1)}{3 - 2} = \frac{2 + 1}{1} = 3 $$

Step 3: Find the y-intercept (\( b \))

Now that we know \( m = 3 \), our function looks like:

$$ f(x) = 3x + b $$

To find \( b \), substitute one of the given points into this equation. Let's use \( f(3) = 2 \):

$$ 2 = 3(3) + b $$

$$ 2 = 9 + b $$

$$ b = 2 - 9 $$

$$ b = -7 $$

Step 4: Write the final algebraic expression

Substitute \( m = 3 \) and \( b = -7 \) back into the general form:

$$ f(x) = 3x - 7 $$

Answer:

$$ f(x) = 3x - 7 $$