What is a Function?

To begin, let’s define what a function is. In mathematics or algebra, a function is an expression that specifies the relationship between two specific variables, or, to put it another way, a function is a relationship between a system’s input and output. Functions determine the output value based on the value of the input. It’s a simplified relationship between distinct words’ values. As an example,

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Let y = x + 7

Then, it can also be written as, y = f(x) i.e., function of x = x+ 7.

Now, for any value of x, the value of y will be different.

For x = 3;

Y = f(3) = 3 + 7 = 10.

Let’s look at the idea of a function’s domain and range. Let’s speak about a function’s domain. A function’s domain can be defined as all of the function’s potential values that can be utilized as input. Consider the following function to have a better understanding.

f(x) =  – 3

We can easily see in the following code that any genuine value of ‘x’ can provide a legitimate result. As a result, the domain of this function is all real numbers, i.e., x R. There are just a few facts that may be used to determine the domain.

1. All values of ‘x’ for which the output is valid can be included in the domain.
2. The denominator of the function can never be zero.
3. If the function is a square root of ‘x’ i.e., f(x) = √x, then the value of x can never be less than zero, otherwise we will get imaginary outputs.

Now let us move further and have a look at how to determine the range of a function; we have to follow the following steps one by one-

1. Put y = f(x).
2. Express ‘x’ as a function of ‘y’.
3. Note all the probable values for ‘y’.
4. Eliminate values by looking into the initial function.

To understand this, let us consider a function, f(x) = (x-2) / (3-x) for all real values of x.

Now we will proceed by using he above steps,

Y = f(x) = (x-2) / (3-x)

→y = (x-2) / (3-x)

→x – 2 = 3y – xy

→x + xy = 3y + 2

→x = (3y + 2) / (y + 1)

From the above expression, we can conclude that all the values of ‘x’ will be acceptable accept when ‘y = (-1)’ as the value of ‘x’ will be undefined. Hence, we can conclude that the range of this function will be = R (real numbers) – {-1} i.e., R – {-1}.

Now we’ll talk about the square root function, which is a significantly important sort of function. A square root function can be defined as a function of ‘x’ such that x =  or x = . To get a clear idea, let us take an example-

f(x) =

The function described above is known as a square root function, and it can never return a negative number, which is an essential point to remember. To determine the domain of this function, we must first determine all of the potential values of ‘x’ for which the entire function is valid. In this case, the function is valid if and only if (x – 5) is greater than zero i.e.

x – 5  0

→x  5

So, the domain of the above function will be; x  (5, ∞) which means ‘x’ can be any value from 5 to infinity excluding 5 and infinity.

For the same function, if we are willing to find out the range, then again, we will have to follow the steps mentioned above,

Consider, y = f(x) =

→ =

→x – 5 = 1 /

→x =  – 5

→ x =

As we have expressed the value of ‘x’ in terms of ‘function of y’, now we can find out the range of the given function. From the above expression, we can clearly conclude that the value of ‘x’ is valid only if the denominator i.e.,  is not equal to zero or y is not equal to zero. So, we can remove ‘0’ from the all the possible real value of ‘y’ which means y ϵ R – {0}.

But as it is a square root function, it has a slightly different characteristic which says that the value of a square root function can never be a negative value. Thus, the value of ‘y’ will also be excluded from the negative part of the real numbers. In simple words, y can never be negative as well as ‘0’. So, the range of the above given function will be;

Y ϵ (0, ∞)

The open interval “()” is used to highlight the fact that the value of ‘y’ is from 0 to ∞, excluding 0 and ∞. So, this is how we determine the range of a square root function.

Can you solve this problem: function f: R to R: f(x) =x^(3) is?

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