Modulus Function and Inverse Graphs: PECTAA Class 12 Maths, Punjab Board

What is |−5|? Five. What is |5|? Also five. The modulus function throws away the sign, and its graph shows that in one picture: a sharp V. The same unit then shows a second picture trick: fold a graph along y = x and you get its inverse.

Unit 1 of the PECTAA Mathematics 12 book is Graphical Representation of Functions. Section 1.7 covers the modulus function and its graph. Section 1.8 links a one-to-one function to its inverse through the graph. This note is for FSc Part 2 (2nd year, Class 12) students in Lahore, Faisalabad, Sargodha, Rawalpindi, Multan, Gujranwala and the other Punjab boards. Gujrat district comes under the Gujranwala board. PECTAA writes the books. Your own board sets your paper.

What to know

The modulus function (section 1.7). It is also called the absolute value function. It is written f(x) = |x| and defined in two pieces:

• |x| = x when x ≥ 0
• |x| = −x when x < 0

So |2| = 2, |−2| = −(−2) = 2 and |0| = 0. The output is never negative.

Its graph. The graph of y = |x| is V-shaped. It is two straight lines that meet at the origin:

• for x ≥ 0 it is the line y = x
• for x < 0 it is the line y = −x

The book sketches it from three points: (0, 0), (2, 2) and (−2, 2). The two arms are mirror images in the y-axis, and the lowest point is the corner at (0, 0).

When an inverse exists (section 1.8). A function has an inverse only if it is bijective, meaning one-to-one and onto. The book adds a useful remark. If the co-domain is chosen equal to the range, every value in the co-domain is already reached, so the function is onto. Then only the one-to-one condition needs checking.

Why the line y = x. A point (x, y) on a function is an input-output pair. The inverse swaps input and output, so (x, y) becomes (y, x). The line y = x is exactly where input equals output, which makes it the natural mirror for this swap.

Method for drawing an inverse graph (section 1.8.1).

1. Draw the graph of the one-to-one function.
2. Draw the line y = x on the same axes.
3. Reflect the graph in the line y = x.
4. The reflected curve is the graph of the inverse.

Each point and its image are the same perpendicular distance from y = x, on opposite sides.

Key points

• |x| ≥ 0 always. The graph never goes below the x-axis.
• y = |x| is not one-to-one on all real numbers, because x and −x give the same output. So it has no inverse there unless the domain is restricted, for example to x ≥ 0.
• Domain and range swap between a function and its inverse. The reflection shows this directly.

Worked examples

Example 1 (book Example 12). Show graphically that y = ln x is the inverse of y = eˣ.

1. y = eˣ is one-to-one on ]−∞, ∞[ and its range is ]0, ∞[. So it is invertible, and the inverse is defined on ]0, ∞[.
2. Draw y = eˣ and the line y = x on the same axes.
3. Reflect points of y = eˣ in y = x. For example (0, 1) goes to (1, 0).
4. The curve through the reflected points is y = ln x, defined on ]0, ∞[.

Example 2. Evaluate |3 − 7| and |−4| + |4|.

1. |3 − 7| = |−4| = 4.
2. |−4| + |4| = 4 + 4 = 8.

Example 3. Is y = |x| one-to-one on all real numbers?

1. Take x = 3 and x = −3.
2. Both give y = 3.
3. Two inputs share one output, so it is not one-to-one, and there is no inverse on that domain.

Example 4. The point (2, 9) lies on a one-to-one function f. Which point lies on the graph of its inverse?

1. Swap the coordinates.
2. The point is (9, 2). It is the mirror image of (2, 9) in y = x.

Common mistakes in the exam

1. Writing |x| = x for every x. For negative x, |x| = −x, which is a positive number.
2. Drawing a U curve for y = |x|. The graph is made of two straight lines with a sharp corner at the origin, not a smooth curve.
3. Reflecting in the x-axis or the y-axis. The inverse comes from reflection in the line y = x.
4. Forgetting the one-to-one check. A function that is not one-to-one has no inverse without restricting its domain.
5. Giving ln x a domain of all real numbers. The book defines it on ]0, ∞[ only.
6. Not swapping domain and range. The domain of eˣ becomes the range of ln x, and the range of eˣ becomes the domain of ln x.

Exam technique

• For the modulus graph, plot (0, 0), (2, 2) and (−2, 2), as the book does, then join with straight lines.
• Always draw y = x as a dashed line before reflecting.
• Mark one pair of matching points, such as (0, 1) and (1, 0), to show the reflection clearly.
• State domain and range for both curves in an inverse question.
• Check your board's notice for the sections in your paper.

Practice questions

1. Find |−9| and |0|. Answer: 9 and 0.
2. Which line is the graph of y = |x| for x < 0? Answer: y = −x.
3. State the condition for a function to have an inverse. Answer: It must be bijective (one-to-one and onto). With the co-domain taken as the range, one-to-one is enough.
4. The point (−1, 5) is on a one-to-one function. Give the matching point on its inverse. Answer: (5, −1).
5. What is the domain of y = ln x, the inverse of y = eˣ? Answer: ]0, ∞[, the range of eˣ.

Ready to practise this? Continue in 2nd Year Mathematics.