Absolute Value Functions

The absolute value function is explored through definitions, examples and exercises with solutions at the bottom of the page. The graphing of absolute value functions |f(x)| is discussed through examples.

Definition of Absolute Value

Consider the number line and a point at a distance x from the origin zero of the number line.
The absolute value of x written as |x| is the distance from zero to x. Since |x| gives a distance, it is always positive or equal to zero. Hence
| -5 | = 5
| 5 | = 5
| 0 | = 0
|a - b| is the distance between points at a and b on the number line.
|5 - 15| = | - 10| = 10
definition
\[ |x| = \left\{ \begin{array}{ll} x & x\geq 0 \\ -x & x\lt 0 \\ \end{array} \right. \]
which can be used to write
\[ |x - a| = \left\{ \begin{array}{ll} x - a & x\geq a \\ - (x - a) = a - x & x\lt a \\ \end{array} \right. \]

Absolute Value and Square Root

\( \sqrt { x^2} = | x | \) ; because the square root is positive or equal zero.
\( \sqrt {(x - a)^2} = |x - a| \)

Properties of the Absolute Value Function

  1. \( | x | \ge 0 \)
  2. \( \sqrt{ x^2} = | x | \)
  3. \( | x | = | - x | \)
  4. \( | a b | = | a || b | \)
  5. \( \left | \dfrac{a}{b} \right | = \dfrac{|a|}{|b|} , b \ne 0\)

Equations and Inequalities with Absolute Value

For \( k \ge 0 \)
  1. \( | x | = k \iff x = k \text{ or } x = - k \)
  2. \( | x | \le k \iff - k \le x \le k \)
  3. \( | x | ≥ k \iff x \le - k \text{ or } x \ge k \)

Graph of Absolute Value of a Function

Function f(x) used is a quadratic function of the form
\[ f(x) = a x^2 + b x + c \] The exploration is carried out by changing the parameters a, b and c included in \( f(x)\) above.

Interactive Tutorial



a =
-10+10

b =
-10+10

c =
-10+10

>

  1. click on the button above "draw" to start.
  2. Use the sliders to set parameter a to zero, parameter b to zero and parameter c to a positive value; \( f(x)\) is a constant function. Compare the graph of f(x) in blue and that of \( h(x)= |f(x)| \) in red. Change c to a negative value and compare the graphs again. Use the definition of the absolute value functions to explain how can the graph of \( |f(x)| \) be obtained from the graph of \( f(x)\).
  3. Keep the value of a equal to zero, select non zero values for b to obtain a linear function . How can the graph of \( h(x) = |f(x)|\) be obtained from that of \( f(x)\)?
    Hint: use the definition of the absolute value functions and reflection of a graph on the x-axis.
  4. Set b and c to zero and select a positive value for a to obtain a quadratic function . Why are the two graphs the same? (Hint: use the definition of the absolute value functions).
  5. Set b and c to zero and select a negative value for a to obtain a quadratic function . Why are the two graphs reflection of each other? (Hint: use the definition of the absolute value functions and reflection of a graph on the x-axis).
  6. Keep the values of a and b as in 5 above and change gradually c from zero to some positive values. How can the graph of \( h(x)=|f(x)|\) be obtained from that of \( f(x)\)?
  7. Select different values for a, b and c and explore.

Exercises

Part 1
Evaluate the expressions
  1. | - 9 |
  2. | 9 |
  3. | 2 - 6 |
  4. \( |- \dfrac{5}{2}| \)
  5. \( | 11 - 4^2 | \)
  6. \( 2^{| 2 - 7|} \)
  7. \( |2-19| - |9 - 20| \)
Part 2
Substitute and Evaluate the expressions
  1. | x - 10 | + | 2 x | , x = - 7
  2. | - x | , x = 8
  3. \( \left |- \dfrac{|x - 2|}{|-x + 9|} \right | \) , x = -20
  4. \( | - 3 x - x^2 | \) , x = -3
  5. \( x^{| x - 7|} \) , x = - 2
Part 3
Simplify the expressions
  1. | a - b | for a = b
  2. | a - b | for a > b
  3. | a - b | for b > a
  4. \( \sqrt {3^2} \)
  5. \( \sqrt {(- 3)^2} \)
  6. \( \sqrt { (12 - 20)^2 } \)
  7. \( | x^2 | \)
  8. \( | (-x)^2 | \)
  9. \( | (x + 1)^2 | \)
  10. \( \sqrt{(x - y + z)^2} \)
  11. \( \sqrt{sin^2(x)} \)
Part 4
Solve the Equations
  1. \( |x - 2| = 4 \)
  2. \( |x + 5| = 0 \)
  3. \( |10 - x| = - 5 \)
Part 5
Solve the Inequalities
  1. \( |x - 7| \gt 4 \)
  2. \( |x + 5| \lt 9 \)
  3. \( |x + 8| \lt - 5 \)
  4. \( |x + 8| \gt - 2 \)
Part 6
Sketch the following pairs of functions in the same system of axes and explains the similarities and differences of the two graphs.
  1. \( f(x) = x - 1 , h(x) = |f(x)| \)
  2. \( f(x) = x^2 - 4 , h(x) = |f(x)| \)
  3. \( f(x) = \sin(x) , h(x) = |\sin(x)| \)

Solutions to the Above Exercises
Part 1
Evaluate the expressions

  1. | - 9 | = 9
  2. | 9 | = 9
  3. | 2 - 6 | = 4
  4. \( |- \dfrac{5}{2}| = \dfrac{5}{2} \)
  5. \( | 11 - 4^2 | = 5 \)
  6. \( 2^{| 2 - 7|} = 32 \)
  7. \( |2-19| - |9 - 20| = 6 \)
Part 2
Substitute and Evaluate the expressions
  1. For x = - 7, | x - 10 | + | 2 x | = 31
  2. For x = 8 , | - x | = 8
  3. For x = - 20 , \( \left |- \dfrac{|x - 2|}{|-x + 9|} \right | = \dfrac{22}{29}\)
  4. For x = - 3 , \( | - 3 x - x^2 | = 0 \)
  5. For x = - 2 , \( x^{| x - 7|} = - 512\)
Part 3
Simplify the expressions
  1. | a - b | = 0
  2. | a - b | = a - b
  3. | a - b | = b - a
  4. \( \sqrt {3^2} = |3| = 3 \)
  5. \( \sqrt {(- 3)^2} = |-3| = 3 \)
  6. \( \sqrt { (12 - 20)^2 = 8} \)
  7. \( | x^2 | = x^2\)
  8. \( | (-x)^2 | = x^2\)
  9. \( | (x + 1)^2 | = (x + 1)^2 \)
  10. \( \sqrt{(x - y + z)^2} = |x - y + z|\)
  11. \( \sqrt{sin^2(x)} = |sin(x)| \)
Part 4
Solve the Equations
  1. \( |x - 2| = 4 \) , two solutions: x = - 2 and x = 6
  2. \( |x + 5| = 0 \) , one solution: x = - 5
  3. \( |10 - x| = - 5 \) , no real solutions.
Part 5
Solve the Inequalities
  1. \( |x - 7| \gt 4 \) , solution set: x < 3 ∪ x > 11
  2. \( |x + 5| \lt 9 \) , solution set: -14 < x < 4
  3. \( |x + 8| \lt - 5 \) , solution set: {∅}
  4. \( |x + 8| \gt - 2 \) , solution set: {all real numbers}
Part 6
  1. \( f(x) = x - 1 , h(x) = |f(x)| = |x - 1| \)
    For x - 1 ≥ 0 or x ≥ 1, h(x) = | x - 1 | = x - 1 and we therefore have h(x) = f(x) which we can be clearly seen from the graphs.
    For x - 1 < 0 or x < 1 , h(x) = |x - 1| = - (x - 1) = - f(x).
    Therefore for x < 1 h(x) = - f(x) which means that the graph of h is a reflection of the graph of f on the x axis and this is can be clearly seen on the graphs.

    graph of f(x) =  x - 1 and h(x) = |x - 1|

  2. \( f(x) = x^2 - 4 , h(x) = |f(x)| = |x^2 - 4| \)
    For x2 - 4 ≤ 0 or -2 ≤ x ≤ 2 , h(x) = |(x2 - 4)| = - (x2 - 4) = - f(x) . Therefore, on the interval -2 ≤ x ≤ 2, the graph of h is a reflection of the graph of f. Outside this interval, the two graphs coincide.

    graph of f(x) =  x<sup>2</sup> - 4 and h(x) = |x<sup>2</sup> - 4|

  3. \( f(x) = \sin(x) , h(x) = |\sin(x)| \)
    Over intervals -2π to -π , 0 to π , 2π to 3π, ... sin(x) ≥ 0 and h(x) = sin(x) = f(x) therefore the graphs of h and f coincide.
    For intervals - π to 0 , π to 2π ... sin(x) < 0 , h(x) = | sin(x) | = - sin(x) and the graph of h is a reflection of the graph of f on the x axis.

    graph of f(x) =  sin(x) and h(x) = |sin(x)|
Conclusion: In all three exercises above, the graph of h is on or above the x axis because h(x) = |f(x)| and the absolute value is positive or equal to 0.

More References and Links

Simplify Absolute Value Expressions
Absolute Value Equations And Inequalities.
Absolute Value.

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