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

Summary

This video explains the concept of the modulus function (also known as absolute value function). The script covers its general form, f(x) = |x|, and demonstrates how to plot its graph through an example. It shows the breakdown of the modulus function based on whether the value inside the absolute value symbol is positive or negative. The presenter goes through specific examples of calculating the modulus function for different values, highlighting the graphical representation and its behavior. The video also touches on whether the function is injective, surjective, or bijective, concluding that the modulus function is surjective but not injective or bijective.

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Q & A

What is a modulus function, and how is it related to absolute value?

A modulus function, also known as an absolute value function, returns the non-negative value of its argument. It can either be positive or zero if the argument is non-negative, or it is the negative of the argument if the argument is negative.

What is the general form of a modulus function?

The general form of a modulus function is f(x) = |x|, where |x| represents the absolute value of x.

How do you define the modulus of a function, and how does it behave?

The modulus function is defined by two cases: if x is greater than or equal to 0, then f(x) = x; if x is less than 0, then f(x) = -x. The result is always non-negative.

What is the importance of the 'breakpoint' in graphing a modulus function?

The 'breakpoint' occurs at the value where the argument inside the modulus changes sign. In the given example, this occurs at x = 3. The graph will reflect different behaviors on either side of this breakpoint.

How do you graph a modulus function?

To graph a modulus function, first identify the breakpoint where the argument inside the modulus changes sign. Then, for values greater than or equal to the breakpoint, the function behaves linearly as f(x) = x - 3, and for values less than the breakpoint, the function behaves as f(x) = -(x - 3). Plot these points and connect them to form the graph.

What are the key points to plot when graphing the modulus function f(x) = |x - 3|?

Key points to plot include the breakpoint at (3, 0), and points on either side of this, such as (-2, 5), (-1, 4), (0, 3), (1, 2), and so on, based on the values of x and f(x).

What does it mean for a function to be injective, and is the modulus function injective?

An injective function is one where every element of the domain maps to a unique element in the codomain. The modulus function is not injective because multiple values of x can produce the same output. For example, f(-2) = f(2) = 5.

What is a surjective function, and is the modulus function surjective?

A surjective function is one where every element of the codomain has a corresponding element in the domain. The modulus function is surjective when its range is non-negative real numbers, as every non-negative number has a corresponding input in the domain.

Why is the modulus function not bijective?

The modulus function is not bijective because it is not injective. For a function to be bijective, it must be both injective and surjective, but the modulus function fails the injectivity condition.

What is the domain and range of the modulus function f(x) = |x - 3|?

The domain of the modulus function is all real numbers, as the function is defined for any real value of x. The range is all non-negative real numbers, as the output of the function is always zero or positive.