The graph of $h(x)=|x-10|+6$ is shown. On which interval is this graph increasing?
A. $(-\infty, 6)$
B. $(-\infty, 10)$
C. $(6, \infty)$
D. $(10, \infty)$
Answer (2)
Rewrite the absolute value function as a piecewise function.
Determine the slope of the function for x < 10 and 10"> x > 10 .
Identify the interval where the slope is positive, indicating an increasing function.
The function is increasing on the interval ( 10 , ∞ ) .
Explanation
Understanding the Problem We are given the function h ( x ) = ∣ x − 10∣ + 6 and asked to find the interval on which the graph of this function is increasing.
Rewriting as a Piecewise Function The absolute value function can be written as a piecewise function. When x < 10 , we have x − 10 < 0 , so ∣ x − 10∣ = − ( x − 10 ) = 10 − x . Thus, h ( x ) = 10 − x + 6 = 16 − x . When xg e 10 , we have x − 10 g e 0 , so ∣ x − 10∣ = x − 10 . Thus, h ( x ) = x − 10 + 6 = x − 4 .
Analyzing the Intervals So, we can write h ( x ) as h ( x ) = { 16 − x x − 4 if x < 10 if x ≥ 10 When x < 10 , the slope of the line is − 1 , so the function is decreasing. When 10"> x > 10 , the slope of the line is 1 , so the function is increasing.
Determining the Increasing Interval Therefore, the function is increasing on the interval ( 10 , ∞ ) .
Examples
Understanding increasing and decreasing intervals of functions is crucial in many real-world applications. For instance, consider a business tracking its profit over time. If the profit function is increasing on a certain interval, it indicates a period of growth. Similarly, in physics, analyzing the velocity function of an object can reveal when the object is accelerating (increasing velocity) or decelerating (decreasing velocity). These concepts are fundamental in economics, engineering, and various scientific fields.
The function h ( x ) = ∣ x − 10∣ + 6 is increasing for values of x greater than 10. Therefore, the correct answer is that the graph is increasing on the interval ( 10 , ∞ ) . Hence, the answer is Option D.
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