Fungsi #Part 13 // Jenis-jenis Fungsi // Fungsi Modulus // Fungsi Mutlak // Grafik, Domain , Range

Summary

This video provides an in-depth explanation of the modulus (absolute value) function, including its mathematical definition, how to graph it, and its key properties. The modulus function is broken down into two cases: one for positive inputs and one for negative inputs. The speaker demonstrates plotting the function step-by-step, showing its 'V' shape with a vertex at (3, 0). Additionally, the video discusses the function's domain, range, and its non-injective yet surjective nature, ultimately concluding that it is not bijective. This tutorial is helpful for understanding both the theory and application of modulus functions.

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Q & A

What is the modulus function?

The modulus function, also known as the absolute value function, outputs the non-negative magnitude of a number, regardless of its sign. Mathematically, for any number x, the modulus function is defined as |x|, where: if x ≥ 0, |x| = x, and if x < 0, |x| = -x.

What is the general form of the modulus function?

The general form of the modulus function is f(x) = |x|. This function takes an input value x and returns the absolute value, which is always non-negative.

How does the modulus function affect negative values?

For negative values of x, the modulus function changes the sign of the input. For example, if x = -5, then |x| = 5. Essentially, any negative input is reflected to its positive equivalent.

What is the graph of the function f(x) = |x - 3|?

The graph of f(x) = |x - 3| is a 'V' shaped curve with its vertex at the point (3, 0). The function has a slope of 1 for x ≥ 3 and a slope of -1 for x < 3, reflecting the behavior of the absolute value function.

What is the significance of the point x = 3 in the function f(x) = |x - 3|?

The point x = 3 is the vertex of the graph of f(x) = |x - 3|. This is where the function changes direction, and the value of the function at this point is zero, i.e., f(3) = 0.

How do you plot the function f(x) = |x - 3|?

To plot f(x) = |x - 3|, first determine the key points such as f(-2), f(-1), f(0), f(1), etc., based on the modulus formula. Then, plot these points on the Cartesian plane and connect them. The graph will have a 'V' shape with a vertex at (3, 0).

What are some example values for f(x) = |x - 3|?

Example values for the function f(x) = |x - 3| include: f(-2) = 5, f(-1) = 4, f(0) = 3, f(1) = 2, f(2) = 1, f(3) = 0, f(4) = 1, f(5) = 2, f(6) = 3.

What does the 'V' shape of the modulus function graph signify?

The 'V' shape of the modulus function graph signifies that the function has a turning point (the vertex) where the value changes direction. The slope is positive on one side of the vertex and negative on the other, reflecting the absolute value operation.

Is the modulus function injective (one-to-one)? Why or why not?

No, the modulus function is not injective because it does not map distinct inputs to distinct outputs. For example, f(-2) = f(4) = 5, meaning that different input values can yield the same output.

Is the modulus function surjective (onto)?

Yes, the modulus function is surjective because every non-negative value is covered in its range. In the case of f(x) = |x - 3|, the range is [0, ∞), meaning all non-negative real numbers are possible outputs.