Domain And Range Of Log

Finding The Domain And Range Of A Function: Overview, Method, Examples

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Finding the Domain and Range of a Part: Domain, in mathematics, is referred to as a whole set up of imaginable values. These values are independent variables. In other words, in a domain, we have all the possible x-values that will brand the function piece of work and will produce real y-values. The range, on the other hand, is set as the whole prepare of possible yielding values of the depending variable, which in this case, is y (generally).

Finding the domain requires determining the values of the independent variables (which is usually x) that have been allowed to utilise. At the bottom of the fraction, 0 is usually debarred or we generally avoid negative values that are establish under the square root sign. The range of a function is considered as an assortment of possible y-values. Continue reading to learn more about the domain and range of a function.

Definition of a Office

The relation \(f\) from set \(A\) to fix \(B\) is a function if every element of set \(A\) has only one image in set \(B\). It is a subset of \(A \times B\). Here, the relation \(R\) is a function from the gear up \(A\) to \(B\).

What is the Domain, Codomain, and Range of a Part?

The set up of elements in \(A\) that are plugged into the function \(f\) is called the domain.

The set \(B\) that is a collection of possible outcomes is chosen the codomain. The set of images of the elements in \(A\), which is a subset of \(B\) is called the range of the office \(f\)

Range \(\left\{ {y \in Y,y = f\left( x \right),\,x \in X} \right\}\)

For the function \(R:\)

Domain \( = ~\left\{ {1,~ii,~3} \correct\}\)

Codomain \( = ~\left\{ {five,~vi,~7,~8} \right\}\)

Range \( = ~\left\{ {5,~vi,~8} \correct\}\)

How to Find the Domain of a Office?

The domain of a part is the values for which the role is defined.

For real-valued functions: first, you demand to identify the values for which the function is not defined and then exclude them.

Instance: A logarithmic role \(f(x)=\log x\) is defined only for positive values of \(x\). That is, the domain of the function is the set up of positive real numbers. So, that is how it, i.east., domain and range of logarithmic functions, works.

How to Notice the Range of a Function?

The range is the set of images of the elements in the domain.

To discover the range of a function:

Step 1: Write downwardly the function in the form \(y=f(x)\)
Step two: Solve it for \(x\) to write information technology in the class, \(x=yard(y)\)
Step 3: The domain of the role \(g(y)\) is the range of \(f(x)\).

Case: For the office \(f(x)=\log x\), the image takes up the values from \(-\infty\) to \(+\infty\). That is, the range of the function is the fix of all real numbers.

Notice the Domain and Range from Graphs

We know that the domain of a function is the set up of all input values. So, the domain on a graph is all the input values shown on the \(x\)-axis. To find the domain, we need to analyse what the graph looks similar horizontally. Moving from left to right forth the \(ten\)-axis, identify the span of values for which the function is defined.

Similarly, the range of a function is the set of all output values. On a graph, this is identified every bit the values that are taken by the dependent variable \(y\). So, to find the range, expect at the set of values that the graph spreads vertically. Those looking for the domain and range figurer should take help from the figures shown on this page.

Consider the graph of the function \(y=\sin x\).

Looking at the horizontal and vertical spread of the graph, the domain, and the range tin can be calculated every bit shown below.

The closed points on either end of the graph indicate that they are besides part of the graph. Therefore, the domain is \( – \pi \le ten \le \pi ,\) and the range is \(-1 \leq y \leq 1\).

Now, if you take open points instead, the role is non divers at that point.

Here, the domain is \( – \pi \le x < \pi ,\) and the range is \(-1 \leq y \leq 1\). Annotation that, here the value \(y=0\) is included in the range as it already has a pre-image at \(x = \, – \pi \) and \(ten=0.\)

Sometimes the graph continues beyond the portion shown. In such cases, the domain and range could be greater than the visible values.

Generally, the arrows on either end show that the graph extends infinitely in both directions, and hence, the domain is the set of all real numbers. Still, the range in this particular case remains the same equally \(-ane \leq y \leq 1\)

Find the Domain and Range from Equations

Allow \(y=f(x)\) be the function we need to detect the domain and the range.

Stride ane: Solve the equation to determine the values of the independent variable \(10\) and obtain the domain.

Step 2: To calculate the range, rewrite the equation \(y=f(x)\) with the independent variable \(x\) expressed in terms of \(y\). That is, in the grade \(x=g(y)\). Now, the domain of the function \(k(y)\) is the range of the office \(f(x)\).

Find the Domain and Range of Special Functions

Rational Function: A rational function is divers for but the non-zilch values of the denominator.

Equate the denominator to zero and solve for \(x\) to find the values to be excluded.
Once the values are excluded in the domain, the range is calculated past excluding the images of those values.

Square Root Function: A square root function is defined for merely the not-negative values of the expression nether the radical symbol.

Observe the excluded values for \(x\).
The range is calculated by omitting the images of the excluded values in the domain.

Solved Examples

Q.ane. A function \(f(x)=3 x\) is divers from set \(A\) to set up \(B\) where \(A = ~\left\{{1,~two,~three,~4,~5} \correct\}\) and \(B = ~\left\{{0,~ane,~2,~3,~4,~5,~half dozen,~7,~8,~nine,~x,~11,~12,~13,~14,~15,~sixteen}\right\}\). What are the domain and range of the role \(f\) ? Are range and codomain the aforementioned?
Ans:
Domain, \(A = ~\left\{{1,~2,~3,~4,~five} \correct\}\)
Codomain, \(B = ~\left\{{0,~one,~two,~3,~4,~5,~6,~7,~8,~9,~10,~11,~12,~13,~fourteen,~15,~xvi}\right\}\)
Range is the set of all \(f(x)\)'due south for every \(x∈A\)

\(f(1)=iii\)

\(f(2)=half-dozen\)

\(f(3)=ix\)

\(f(4)=12\)

\(f(5)=15\)

\(\therefore\) Range, \(B = ~\left\{{3,~six,~9,~12,~fifteen} \right\}\)
Hence, range \(\neq\) codomain.

Q.2. Find the domain of \(f(x)=\frac{x^{2}+two x+i}{ten^{2}+3 x+two}\).
Ans: Given: \(f(x)=\frac{x^{2}+2 x+1}{10^{2}+3 10+ii}\)
\(=\frac{(x+one)^{2}}{(x+1)(ten+2)}\)
\(=\frac{10+1}{ten+2}\)
Since a rational role is defined just for non-zero values of its denominator, we have,
\(x+2 \neq 0\)
\(\Rightarrow x \neq-ii\)
\(\therefore\) Domain \( = \left\{ {10 \in R,10 \ne – 2} \correct\}\)

Q.3. What are the domain and range of the real-valued function \(g(x)=\sqrt{75-ten^{2}}\) ?
Ans: A foursquare root role is defined only for non-negative values under the square root symbol.
Given: \(75-x^{two} \geq 0\)
\(\Rightarrow 75 \geq x^{2}\) or \(x^{ii} \leq 75\)
\(-\sqrt{75} \leq x \leq \sqrt{75}\)
\(\therefore\) Domain \(={x \in R,-\sqrt{75} \leq x \leq \sqrt{75}}\) or \([-\sqrt{75}, \sqrt{75}]\)
Permit \(y=\sqrt{75-x^{two}}\)
\(y^{ii}=75-10^{2}\)
\(x^{2}=75-y^{2}\)
Since \(x \in[-\sqrt{75}, \sqrt{75}]\), the value of \(y\) varies from \(0\) to \(\sqrt{75}\)
\(\therefore\) Range \(={y \in R, 0 \leq y \leq \sqrt{75}}\) or \([0, \sqrt{75}]\)

Q.4. Find the domain for which the functions \(f(x)=2 x^{2}+3 10+1\) and \(g(ten)=ten^{2}-v x-14\) are equal.
Ans:
Given: \(f(x)=g(ten)\)
\(\therefore two x^{ii}+3 x+1=x^{2}-5 x-14\)
\(two x^{two}+3 x+1-10^{2}+5 x+14=0\)
\(ten^{2}+8 x+15=0\)
\((x+3)(x+v)=0\)
\(x=-three\) or \(10=-5\)
Therefore, the domain for which the functions \(f(x)\) and \(g(x)\) are equal is \(\left\{ { – 3,\, – v} \right\}\)

Q.5. Identify the domain and range of the office represented past the graph.

Ans: The open up dot on the left extreme shows that the plotted point is non included. That is, the function is not defined for the point \(10=-1\).
From the graph, the function is divers for all the values from \(-1\) to \(3\), including \(iii\) and excluding \(-1\).
So, the domain of the function is \(-i<x \leq 3\)
The range varies from \(0\) to \(4\), including both points.
That is, the range is \(0 \leq y \leq 4\).

Domain: \({10 \in R,-1<x \leq three}\) or \((-one,3]\)
Range: \({x \in R, 0 \leq ten \leq 4}\) or \([0,4]\)

Summary

The article defines a function, its domain, range, and codomain. It goes on to explain each in particular with examples. Farther, it explains the methods to find the domain and range of a function when the function rule or an equation, or the graph of a function is given.

The article besides discusses the central points in finding the domain and range of some special functions such as rational and foursquare root functions. Information technology concludes with a few solved examples to take emphasised the idea of the concepts and the calculations involved. Those searching for the domain and range table should read the article on our website.

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FAQs on Finding the Domain and Range of a Office

Nosotros have added the domain and range examples below then that aspirants can understand unlike aspects of the same and get resolutions to their queries.

Q.1. What is the difference betwixt domain and codomain?
Ans: When a function \(f\) is defined from set \(Ten\) to ready \(Y\):
1. The set of elements in \(Ten\) that are plugged into the function \(f\) is called the domain.
2. The ready \(B\) that is a drove of possible outcomes is called the codomain.

Q.2. How to notice the domain and range of a office algebraically?
Ans: To notice the domain of a role, detect the values for which the function is defined. For instance, a rational function is defined for only the non-zero values of the denominator. So, equate the denominator to zero and solve for \(x\) to find the values to exist excluded.
Now, to find the range of the function, write down the function in the course \(y=f(ten)\) and solve it for \(x\) to write it in the for \(x=g(y)\). At present the domain of the function \(g(y)\) is the range of the part \(f(10)\).

Q.3. How do yous find the domain and range of a office without graphing?
Ans: Let \(f(x)\) be the office to find the domain and the range.
Stride 1: Rewrite the equation representing the function in the class \(y=f(x)\).
Stride 2: Solve the equation to decide the values of the independent variable \(x\) to obtain the domain.
Step 3: Rewrite the equation \(y = f(x)\) with the independent variable 10 expressed in terms of \(y\). That is, in the form \(x=g(y)\).
Pace 4: The domain of the office \(chiliad(y)\) is the range of \(f(x)\).

Q.iv. What is the domain and range in a graph?
Ans: The domain of a function is the set of input values. And so, the domain in a graph is the input values shown on the \(ten\)-axis. The range of a function is the prepare of the output values. On a graph, this can exist identified as the values taken by the dependent variable \(y\). Therefore, on a graph, the domain and range can be found past identifying the range of \(x\) and \(y\)-value variations.

Q.five. How exercise you notice the domain and range of a quadratic equation?
Ans: Consider the parent quadratic function \(f(ten)=x^{2}\).

Observe the graph of the given quadratic equation. Moving from left to right forth the \(ten\)-centrality, at that place are no holes in the graph specifying the points where the function is non defined. So, the domain is the set of all real numbers.

The range of the office is the values that the graph spreads vertically. The \(y\)-values are all greater than or equal to nix. Thus, the range is a prepare of all real numbers greater than or equal to zero.
Since all other quadratic functions are transformations of the parent role, their domain and range can exist calculated equally transformations of this part.

We hope this detailed article on finding the domain and range of a function helped you in your studies. If yous take any doubts, queries or suggestions regarding this article, experience free to ask us in the comment section and nosotros volition exist more than happy to assist y'all. Happy learning!