Find the Range of a Function

A set of numbers that the function can generate. The range of a function can also be a set of values of y by putting all possible values of x into the function. The domain is the set of all possible x-values. You have to go through these steps to understand how to determine the range of a function. In this article, we will tell you how to find the range of a function.

How To Find Range Of A Function

• Jot the formula

Assume that you have to solve an equation that is mentioned below:

 3×2+ 6x -2 = f(x).

This equation implies that you can put any x into the formula and get the value of y. This is a parabola’s function.

• Determine The Vertex Of The Line

If the equation you’re working on is quadratic, then you have to determine the vertex. And you can ignore this part if the equation concerns a straight line or some function with an odd polynomial. For example, the equation mentioned below concerns a straight line. 

 f(x) = 6 x 3+2x + 7. 

But if the given equation concerns a parabola or an x-coordinate is squared or raised to an even power, you must draw a vertex. 

You can use the formula -b/2a, to get the x coordinate of the mentioned equation. 

 3×2 + 6x -2, 

where 3 = a, 6 = b, and -2 = c. -b is -6 in this case, and 2a is 6. 

 So the x-coordinate is -6/6, or -1.

In order to obtain the y-coordinate, enter -1 into the function. f(-1) = 3 (-1)2 + 6 (-1) -2 = 3 – 6 -2 = -5

• Determine Some More Points

Find a few other x-coordinates before you start to find the range of functions. This would help you find sense in the equation. Because it is a parabola and the x2 coordinate is positive, it will point upward.

• Determine The Range Of The Function

Look at the graph’s y-coordinates and determine the lowest point where the graph comes into contact with the y-coordinate. This equation has the lowest y-coordinate, -5, and the graph stretches infinitely above this point. This implies that the function’s range is y = all real numbers -5.   

Solved Question On How To Find Range Of A Function

• Question: Find the domain and range of the function f(x) = -2/x.

Solution: Deduct the denominator from the numerator.

x = 0.

As a result, domain: all real numbers except 0

Except for 0, the range includes all real values of x.

• Question: Determine the domain of the following function.

f(x)  =  (x – 2) / (3 – x)

Solution: y = (x – 2) / (3 – x)

Start by multiplying both sides by (x-3)

y = (3 – x) (x – 2)

x – 2 = 3y – xy

To solve for x, we must first group the x terms.

Add two to both sides.

x = 3y + 2 – xy

On both sides, add xy.

x + xy = 3y + 2

x(1 + y) = 3y + 2

(3y + 2)/(1 + y) = x

We have 1 + y in the denominator; if we substitute -1 for y, the function becomes undefined.

As a result, the range is R  -1.

• Question: Determine the domain of the following function.

f(x)  =  √(16 – x2)

Solution: Taking squares on both sides, we obtain

16 – x2= y2

Now add x2 on both sides

x2 = 16- y2

x=√ (16 – y2)

X will accept any real value if

(16 – y2) ≥ 0  ==> y2 – 16 ≤ 0  ==> (y + 4) (y – 4) ≤ 0

-4 ≤ y ≤ 4  ==>  y ∈ [-4, 4]

Also, y = √(16 – x2) ≥ 0 for all x ∈ [-4, 4].

As a result, y ∈[0, 4] for all x ∈ [-4, 4]

As a result, the range is [0, 4].

• Question: Determine the domain of the following function.

f(x)  =  1 / √(x – 5)

Solution: y = 1 / √(x – 5)

We have x – 5 > 0 for any x greater than 5.

√(x – 5) > 0  ==> 1/√(x – 5) > 0

As a result, f(x) accepts all real values greater than zero.

As a result, the range of f(x) is (0,∞). 

• Question: Find the range. f(x) = √x-1

Solution: Because the function is a square root, it can never produce negative results. As a result, the minimum value at x = 1 can only be 0. As we increase x, the maximum value can reach infinity.

As a result, the function’s range is [0,∞].

• Question: The domain of the function ƒ defined by f(x) = 1/√x-|x|

Solution: Given f(x)= 1/√x-|x|

When choosing a domain set, two things must be kept in mind.

The denominator is never zero. The term that is contained within the square root does not become negative. Let us expand on what is written inside the term within the square root.

In this case, we cannot use either the value x ≥0 or x< 0.

As a result, f is not defined for any x∈R. The domain is an empty set.

Conclusion

A domain defines a set of possible input values whereas a range defines a set of output values in a function. These two are a significant part of maths. This article explains how to find the range of a function and the highest and lowest range.