# How to Find the Range of a Function

Oct 07, 2020 02:57 AM Photo by Kaboompics .com from Pexels.**Tripboba.com** - In mathematics, a function can be understood using the metaphor of a function machine that generates some output in correlation to a given input. That being said if the object x is in the set of inputs (called the domain), then a function f will map the object x to exactly one object f(x) in the set of possible outputs (called the codomain).

The domain of a function is the set of all possible inputs for the function. For example, the domain of f(x)=x² is all real numbers, and the domain of g(x)=1/x is all real numbers except for x=0. Meanwhile, the codomain of a function is the set of its possible outputs.

For example, when we use the function notation f:R→R, we mean that f is a function from the real numbers to the real numbers. On the other hand, range of a function is the set of all output values of a function. For example, when the function f(x) = x2 is given the values x = {1,2,3,...} then the range is {1,4,9,...}.

Functions are very useful in mathematics because they allow us to model real life problems into a mathematical format. So, if you’re wondering how to find the range of a function, here are the tutorials we’ve got for you.

- How to find the domain and range of a function
- How to find the range of a function algebraically
- How to find the range of a rational function
- How to find the range of a quadratic function
- How to find the domain and range of a piecewise function
- How to find the range of a square root function

**How to find the domain and range of a function**

**How to find the range of a function algebraically**

Overall, the steps for algebraically finding the range of a function are:

**Step 1.** Write down y=f(x) and then solve the equation for x, giving something of the form x=g(y).

**Step 2.** Find the domain of g(y), and this will be the range of f(x).

**Step 3.** If you can't seem to solve for x, then try graphing the function to find the range.

**How to find the range of a rational function**

A rational equation is an equation containing at least one fraction whose numerator and denominator are polynomials

One way to know how to find the range of a function of a rational function is by finding the domain of the inverse function. Another way is to sketch the graph and identify the range. Let us again consider the parent function f(x)=1x. We know that the function is not defined when x=0.

**How to find the range of a quadratic function**

A quadratic function is one of the form f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero. The graph of a quadratic function is a curve called a parabola. Parabolas may open upward or downward and vary in "width" or "steepness", but they all have the same basic "U" shape.

To know how to find the domain and range of a quadratic function, for every polynomial function (such as quadratic functions for example), the domain is all real numbers. if the parabola is opening upwards, i.e. a > 0, the range is y ≥ k; if the parabola is opening downwards, i.e. a < 0, the range is y ≤ k.

**How to find the domain and range of a piecewise function**

In mathematics, a piecewise-defined function (also called a piecewise function) is a function which is defined by multiple sub-functions, each sub-function applying to a certain interval of the main function's domain (a sub-domain). It is basically a function that is broken into two or more pieces—each of these pieces has its own parameters.

We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain “boundaries.” For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to $10,000 are taxed at 10%, and any additional income is taxed at 20%. The tax on a total income S would be 0.1S if S≤$10,000 and $1000+0.2(S−$10,000) if S>$10,000.

**How to find the range of a square root function**

To know how to find the range of a function, it must be understood that a square root "y" of a number "x" is a number such as y^2 = x. This definition of the square root function imposes certain restrictions on the domain and range of the function, based on the fact that x cannot be negative.

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