Finding the Range of x for f(x) = x² to be within 0.1 of 9
Objective: Our goal is to find the range of x values where the function f(x) = x² is within 0.1 units of the value 9.
Step 1: Set Up the Inequality
We start with the inequality |x² – 9| < 0.1 to represent that f(x) should be within 0.1 of 9.
Step 2: Remove the Absolute Value
We rewrite the inequality without the absolute value as -0.1 < x² - 9 < 0.1.
Step 3: Add 9 to All Sides
We add 9 to all three sides of the inequality to isolate x². This gives us 8.9 < x² < 9.1.
Step 4: Take the Square Root
We take the square root of all terms to solve for x. This gives us √8.9 < x < √9.1, approximately 2.981 < x < 3.018.
Step 5: Verify with Specific Values
We’ll pick two specific values within the range: x = 2.985 and x = 3.015.
For x = 2.985, f(x) = (2.985)² = 8.91025. The absolute difference |8.91025 – 9| = 0.08975, which is within 0.1.
For x = 3.015, f(x) = (3.015)² = 9.090225. The absolute difference |9.090225 – 9| = 0.090225, which is within 0.1.
Conclusion: The x values must lie between √8.9 and √9.1, or approximately between 2.981 and 3.018, for f(x) = x² to be within 0.1 of 9. The specific values x = 2.985 and x = 3.015 confirm this.