Which graph represents [tex]f(x)=|x-5|+|x-4|[/tex]?
A. graph A
B. graph B
C. graph C
D. graph D
Answer (1)
Analyze the function f ( x ) = ∣ x − 5∣ + ∣ x − 4∣ by considering different intervals for x .
For x < 4 , f ( x ) = 9 − 2 x , a line with a slope of -2.
For 4 ≤ x ≤ 5 , f ( x ) = 1 , a horizontal line at y = 1 .
For 5"> x > 5 , f ( x ) = 2 x − 9 , a line with a slope of 2. The correct answer is D .
Explanation
Analyze the function We are given the function f ( x ) = ∣ x − 5∣ + ∣ x − 4∣ and asked to identify its graph from the options A, B, C, and D. To do this, we will analyze the function by considering different intervals for x .
Case 1: x < 4 Case 1: x < 4 . In this case, ∣ x − 5∣ = 5 − x and ∣ x − 4∣ = 4 − x , so f ( x ) = ( 5 − x ) + ( 4 − x ) = 9 − 2 x . This is a line with a slope of -2 and a y-intercept of 9.
Case 2: 4 <= x <= 5 Case 2: 4 ≤ x ≤ 5 . In this case, ∣ x − 5∣ = 5 − x and ∣ x − 4∣ = x − 4 , so f ( x ) = ( 5 − x ) + ( x − 4 ) = 1 . This is a horizontal line at y = 1 .
Case 3: x > 5 Case 3: 5"> x > 5 . In this case, ∣ x − 5∣ = x − 5 and ∣ x − 4∣ = x − 4 , so f ( x ) = ( x − 5 ) + ( x − 4 ) = 2 x − 9 . This is a line with a slope of 2 and a y-intercept of -9.
Summary of the piecewise function So, f ( x ) is a piecewise linear function: 5 \end{cases}"> f ( x ) = { 9 − 2 x x < 4 1 4 ≤ x ≤ 5 2 x − 9 x > 5 The graph will have a slope of -2 for x < 4 , a constant value of 1 for 4 ≤ x ≤ 5 , and a slope of 2 for 5"> x > 5 .
Evaluate f(4) and f(5) Evaluate f ( 4 ) = ∣4 − 5∣ + ∣4 − 4∣ = ∣ − 1∣ + 0 = 1 + 0 = 1 and f ( 5 ) = ∣5 − 5∣ + ∣5 − 4∣ = 0 + ∣1∣ = 0 + 1 = 1 .
Conclusion Based on the analysis, the graph should have a slope of -2 when x < 4 , be a horizontal line at y = 1 when 4 ≤ x ≤ 5 , and have a slope of 2 when 5"> x > 5 . This description matches graph D.
Examples
Understanding absolute value functions is crucial in various real-world scenarios. For instance, in manufacturing, tolerances are often specified using absolute values. If a machine part is designed to be 5 cm long with a tolerance of ± 0.1 cm, the actual length x must satisfy ∣ x − 5∣ ≤ 0.1 . Similarly, in finance, the absolute value can represent the risk associated with an investment, measuring the deviation from the expected return. Piecewise functions, like the one we analyzed, are used to model situations with different behaviors under different conditions, such as pricing strategies that change based on demand.
