TLDR;
This video provides a comprehensive guide on expressing absolute value functions into several different functions with their limits. It covers examples with quadratic forms, two absolute value signs, and three absolute value signs, explaining how to break down each function and determine the appropriate limits for each case. The key points include understanding the basic principle of absolute values, solving quadratic inequalities, and carefully considering the signs and limits when multiple absolute values are involved.
- Expressing absolute value functions into several different functions with their limits.
- Solving quadratic inequalities.
- Handling multiple absolute values by breaking them down and considering signs and limits.
Contoh Soal 5: Menjabarkan fungsi nilai mutlak yang mengandung bentuk kuadrat menjadi beberapa fungsi dengan batas-batasnya [0:32]
The video explains how to express the function ( f(x) = |4 - x^2| ) into several functions with their respective limits. It starts by revisiting the principle that ( |x| = x ) if ( x \geq 0 ) and ( |x| = -x ) if ( x < 0 ). Applying this, ( |4 - x^2| = 4 - x^2 ) if ( 4 - x^2 \geq 0 ). To find the limits, the quadratic inequality ( 4 - x^2 > 0 ) is solved, which is rewritten as ( x^2 - 4 < 0 ) to make factoring easier. Factoring gives ( (x - 2)(x + 2) < 0 ), leading to critical points ( x = 2 ) and ( x = -2 ). A number line is used to determine the intervals where the inequality holds, resulting in ( -2 \leq x \leq 2 ). Conversely, ( |4 - x^2| = -(4 - x^2) = x^2 - 4 ) if ( 4 - x^2 < 0 ), which leads to ( x < -2 ) or ( x > 2 ). Thus, the function ( f(x) ) is expressed as two separate functions with their defined limits.
Contoh Soal 6: Menjabarkan fungsi nilai mutlak yang mengandung dua tanda mutlak menjadi beberapa fungsi dengan batas-batasnya [4:56]
This section addresses how to express a function with two absolute value signs, ( f(x) = |6 - x| + |2x| - 1 ), into several functions with their limits. The presence of two absolute values means the function will break twice, creating two limits. Each absolute value function is broken down separately. First, ( |6 - x| = 6 - x ) if ( 6 - x \geq 0 ), which simplifies to ( x \leq 6 ), and ( |6 - x| = -(6 - x) = x - 6 ) if ( x > 6 ). Second, ( |2x| = 2x ) if ( 2x \geq 0 ), which simplifies to ( x \geq 0 ), and ( |2x| = -2x ) if ( x < 0 ). A number line is then used to map these conditions, with critical points at ( x = 0 ) and ( x = 6 ), dividing the line into three areas. The signs of each absolute value function are determined for each area, and the original function ( f(x) ) is rewritten for each area by substituting the appropriate expressions for the absolute values. This results in three different functions, each defined over a specific interval.
Contoh Soal 7: Menjabarkan fungsi nilai mutlak yang mengandung tiga tanda mutlak menjadi beberapa fungsi dengan batas-batasnya [12:32]
The video explains how to express a function with three absolute value signs, ( f(x) = |x| + |x + 1| - |2 - x| ), into several functions with their limits. The presence of three absolute values means the function breaks three times, creating four regions. Each absolute value function is broken down: ( |x| = x ) if ( x \geq 0 ) and ( |x| = -x ) if ( x < 0 ); ( |x + 1| = x + 1 ) if ( x + 1 \geq 0 ) (or ( x \geq -1 )) and ( |x + 1| = -(x + 1) = -x - 1 ) if ( x < -1 ); and ( |2 - x| = 2 - x ) if ( 2 - x \geq 0 ) (or ( x \leq 2 )) and ( |2 - x| = -(2 - x) = x - 2 ) if ( x > 2 ). A number line is used with critical points at ( x = -1 ), ( x = 0 ), and ( x = 2 ), dividing it into four regions. The signs of each absolute value function are determined for each region, and the original function ( f(x) ) is rewritten for each region by substituting the appropriate expressions for the absolute values. This results in four different functions, each defined over a specific interval.