Question

question 13. version 1*/1. score: 1/1

let (f(x) = \frac{1}{x + 8}).

(f^{-1}(x) = \frac{1}{x} - 8)

question 14. version 1*/1. score: 1/1

solve the following inequality and graph the solution:

((x - 9)(x - 3) \le 0)

Explanation:

🆕 New Concept Discovered: Solving Quadratic Inequalities
Finding where a parabola is below or on the x-axis.

Step 1: Find the boundary points

Set the quadratic expression equal to zero to find the critical boundary points:

$$ (x - 9)(x - 3) = 0 $$

This gives:

$$ x = 9 \quad \text{or} \quad x = 3 $$

Step 2: Test the intervals

The boundary points \( x = 3 \) and \( x = 9 \) divide the number line into three intervals: \( (-\infty, 3) \), \( (3, 9) \), and \( (9, \infty) \).

We test a value from each interval in the inequality \( (x - 9)(x - 3) \leq 0 \):

• For \( (-\infty, 3) \): Test \( x = 0 \)

$$ (0 - 9)(0 - 3) = (-9)(-3) = 27 \quad (\text{Positive, } > 0) $$

• For \( (3, 9) \): Test \( x = 5 \)

$$ (5 - 9)(5 - 3) = (-4)(2) = -8 \quad (\text{Negative, } \leq 0) $$

• For \( (9, \infty) \): Test \( x = 10 \)

$$ (10 - 9)(10 - 3) = (1)(7) = 7 \quad (\text{Positive, } > 0) $$

The inequality is satisfied in the middle interval where the product is negative or zero.

Step 3: Write the solution in interval notation

Since the inequality is \( \leq \) (less than or equal to), we include the endpoints \( 3 \) and \( 9 \) using square brackets:

$$ [3, 9] $$

Step 4: Graph the solution

To graph this on a number line:

• Place solid dots (or brackets) at \( x = 3 \) and \( x = 9 \) to show they are included.
• Shade the line segment between \( 3 \) and \( 9 \).

       ●===================●
--+----+----+----+----+----+----+----+----+----+--
  0    1    2    3    4    5    6    7    8    9   10

Answer:

Interval Notation:

$$ [3, 9] $$

Graph:
A shaded segment on the number line from \( 3 \) to \( 9 \) with solid endpoints at both \( 3 \) and \( 9 \).