Consider the function. \[ f(x)=|x-6| \] Select all the true statements. The 2 -interoept of the function is \( -\mathbf{0} \). The yintercept of the function is \( \mathbf{6} \). The domain of the function is \( \{x \mid x \geq 6\} \). The function is negative in the interval \( -\infty

Ask by Mccoy Reid. in the United States

Feb 03,2025

Let's analyze each statement for the function \( f(x) = |x - 6| \): 1. **The 2-interoept of the function is \( -\mathbf{0} \).** - **Interpretation:** It seems there might be a typo, possibly referring to the **x-intercept**. The x-intercept occurs where \( f(x) = 0 \): \[ |x - 6| = 0 \implies x = 6 \] So, the x-intercept is at \( x = 6 \), not \( -0 \) (which is essentially \( 0 \)). - **Conclusion:** **False** 2. **The y-intercept of the function is \( \mathbf{6} \).** - **Calculation:** The y-intercept occurs where \( x = 0 \): \[ f(0) = |0 - 6| = 6 \] - **Conclusion:** **True** 3. **The domain of the function is \( \{x \mid x \geq 6\} \).** - **Analysis:** The absolute value function \( |x - 6| \) is defined for all real numbers. - **Conclusion:** **False** 4. **The function is negative in the interval \( -\infty < x < \infty \).** - **Analysis:** The absolute value \( |x - 6| \) is always non-negative (i.e., \( \geq 0 \)) for all real numbers. - **Conclusion:** **False** 5. **The function is decreasing in the interval \( -\infty < x < 0 \).** - **Analysis:** For \( x < 6 \), \( f(x) = 6 - x \), which has a negative slope, meaning the function is **decreasing** as \( x \) increases in this interval. Specifically, in \( -\infty < x < 0 \), the function continues to decrease. - **Conclusion:** **True** **Summary of True Statements:** - The y-intercept of the function is \( 6 \). - The function is decreasing in the interval \( -\infty < x < 0 \). **Final Answer:** All statements except the second and fifth are false. The y-intercept is 6 and the function decreases for all real numbers below zero.