document.write( "Question 1178643: Functions f(x) =1-|x-1| and g(x)=(2x-a) are given .prove that for every 'a' belong to (1,2).the area bounded by the graphs of the given function is less than 1/3.(prove without differentiation and integrals). \n" ); document.write( "
Algebra.Com's Answer #850317 by CPhill(1959)\"\" \"About 
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This is a challenging problem that requires geometric reasoning and careful consideration of the function graphs. Let's break down the proof:\r
\n" ); document.write( "\n" ); document.write( "**1. Understand the Graphs:**\r
\n" ); document.write( "\n" ); document.write( "* **f(x) = 1 - |x - 1|:**
\n" ); document.write( " * This is an absolute value function, creating a \"V\" shape.
\n" ); document.write( " * The vertex of the \"V\" is at (1, 1).
\n" ); document.write( " * The graph intersects the x-axis at x = 0 and x = 2.
\n" ); document.write( " * The graph is symmetric about the line x = 1.
\n" ); document.write( "* **g(x) = 2x - a:**
\n" ); document.write( " * This is a linear function with a slope of 2.
\n" ); document.write( " * The y-intercept is -a.
\n" ); document.write( " * Since a ∈ (1, 2), the y-intercept is between -2 and -1.\r
\n" ); document.write( "\n" ); document.write( "**2. Visualize the Bounded Area:**\r
\n" ); document.write( "\n" ); document.write( "* The graphs of f(x) and g(x) will intersect at two points, creating a bounded area.
\n" ); document.write( "* The shape of this bounded area will be a triangle.\r
\n" ); document.write( "\n" ); document.write( "**3. Find the Intersection Points:**\r
\n" ); document.write( "\n" ); document.write( "* To find the intersection points, set f(x) = g(x):
\n" ); document.write( " * 1 - |x - 1| = 2x - a
\n" ); document.write( "* We need to consider two cases for the absolute value:
\n" ); document.write( " * **Case 1: x ≥ 1**
\n" ); document.write( " * 1 - (x - 1) = 2x - a
\n" ); document.write( " * 2 - x = 2x - a
\n" ); document.write( " * 3x = 2 + a
\n" ); document.write( " * x = (2 + a) / 3
\n" ); document.write( " * **Case 2: x < 1**
\n" ); document.write( " * 1 - (1-x) = 2x -a
\n" ); document.write( " * x = 2x -a
\n" ); document.write( " * x = a
\n" ); document.write( "* Because a is within the interval (1,2) then the intersection points are x = a and x = (2+a)/3.\r
\n" ); document.write( "\n" ); document.write( "**4. Geometric Approach:**\r
\n" ); document.write( "\n" ); document.write( "* **Triangle Formation:** The bounded area is a triangle.
\n" ); document.write( "* **Base of the Triangle:** The base of the triangle is the distance between the two intersection points:
\n" ); document.write( " * Base = |(2 + a) / 3 - a| = |(2 - 2a) / 3| = (2 - 2a) / 3 (since a<2).
\n" ); document.write( "* **Height of the Triangle:**
\n" ); document.write( " * The height is the vertical distance from the vertex of f(x) (1, 1) to the line g(x).
\n" ); document.write( " * The x-coordinate of the vertex of f(x) is x=1.
\n" ); document.write( " * The y-coordinate of g(x) at x=1 is g(1)=2-a.
\n" ); document.write( " * The Height is 1-(2-a) = a-1
\n" ); document.write( "* **Area of the Triangle:**
\n" ); document.write( " * Area = (1/2) * Base * Height
\n" ); document.write( " * Area = (1/2) * [(2 - 2a) / 3] * (a - 1)
\n" ); document.write( " * Area = (1/6) * (2 - 2a) * (a - 1)
\n" ); document.write( " * Area = (-1/3) * (a - 1) * (a - 1)
\n" ); document.write( " * Area = (-1/3) * (a - 1)^2
\n" ); document.write( " * Area = (1/3) * (1-a)^2\r
\n" ); document.write( "\n" ); document.write( "**5. Prove Area < 1/3:**\r
\n" ); document.write( "\n" ); document.write( "* Since a ∈ (1, 2), (1 - a) is a negative value.
\n" ); document.write( "* Therefore (1-a)^2 is a positive value.
\n" ); document.write( "* Since a is between 1 and 2 then 0 < (a-1) < 1.
\n" ); document.write( "* Therefore 0 < (1-a)^2 < 1.
\n" ); document.write( "* Therefore 0 < (1/3)*(1-a)^2 < 1/3.
\n" ); document.write( "* Therefore the area is less than 1/3.\r
\n" ); document.write( "\n" ); document.write( "**Conclusion:**\r
\n" ); document.write( "\n" ); document.write( "For any value of 'a' in the interval (1, 2), the area bounded by the graphs of f(x) and g(x) is less than 1/3. This is proven using geometric properties of the graphs and without the use of differentiation or integration.
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