Questions: Find the horizontal intercepts of the function f(x)=3x-4-1 The intercepts are at x=

Transcript text: Find the horizontal intercepts of the function $f(x)=3|x-4|-1$ The intercepts are at $x=$ $\square$

Solution

Solution Steps

To find the horizontal intercepts (also known as x-intercepts) of the function $ f(x) = 3|x-4| - 1 $, we need to set the function equal to zero and solve for $ x $. This involves solving the equation $ 3|x-4| - 1 = 0 $ for $ x $.

Step 1: Understand the Problem

We need to find the horizontal intercepts (also known as x-intercepts) of the function $ f(x) = 3|x-4| - 1 $. The x-intercepts occur where the function equals zero, i.e., $ f(x) = 0 $.

Step 2: Set the Function Equal to Zero

Set the function equal to zero to find the x-intercepts: \[ 3|x-4| - 1 = 0 \]

Step 3: Solve for the Absolute Value

Add 1 to both sides of the equation: \[ 3|x-4| = 1 \]

Divide both sides by 3: \[ |x-4| = \frac{1}{3} \]

Step 4: Solve the Absolute Value Equation

The equation $ |x-4| = \frac{1}{3} $ implies two cases:

Case 1: $ x-4 = \frac{1}{3} $ \[ x = 4 + \frac{1}{3} = \frac{12}{3} + \frac{1}{3} = \frac{13}{3} \]

Case 2: $ x-4 = -\frac{1}{3} $ \[ x = 4 - \frac{1}{3} = \frac{12}{3} - \frac{1}{3} = \frac{11}{3} \]