Answer
365.4k+ views
Hint: To find the domain of ${a^{x + y}} = {a^x} + {a^y}$, first of all, simplify the function. Then we will take log on both sides and as we know that logarithm of a negative number is not possible, we will take the value of log greater than 0 and find the domain.
Complete step by step solution:
In this question, we are given that y is a function of x and is defined by ${a^{x + y}} = {a^x} + {a^y}$ and we need to find the domain of the function y(x).
Given: ${a^{x + y}} = {a^x} + {a^y}$
First of all, let us see the definition of range.
The domain of a function is the set of all possible input values that produce some output value range.
To find the domain of the given function, we need to simplify it further.
$ \Rightarrow {a^{x + y}} = {a^x} + {a^y}$- - - - - -(1)
Now, we know the property that when two numbers having same base are multiplied, we add their powers. So we can write
${a^{x + y}} = {a^x} \cdot {a^y}$
Therefore, equation (1) becomes
$ \Rightarrow {a^x} \cdot {a^y} = {a^x} + {a^y}$
Now, divide both LHS and RHS with ${a^x} \cdot {a^y}$, we get
\[
\Rightarrow \dfrac{{{a^x} \cdot {a^y}}}{{{a^x} \cdot {a^y}}} = \dfrac{{{a^x} + {a^y}}}{{{a^x} \cdot {a^y}}} \\
\Rightarrow 1 = \dfrac{1}{{{a^y}}} + \dfrac{1}{{{a^x}}} \\
\]
Now, we can write the inverse of any term as $\dfrac{1}{x} = {x^{ - 1}}$. Therefore,
\[
\Rightarrow 1 = {a^{ - x}} + {a^{ - y}} \\
\Rightarrow {a^{ - y}} = 1 - {a^{ - x}} \\
\]
Now, taking log with base a on both sides, we get
\[
\Rightarrow {\log _a}{a^{ - y}} = {\log _a}\left( {1 - {a^{ - x}}} \right) \\
\Rightarrow - y{\log _a}a = {\log _a}\left( {1 - {a^{ - x}}} \right) \\
\]
Now, we know that value of ${\log _a}a = 1$. Therefore, we get
\[
\Rightarrow - y = {\log _a}\left( {1 - {a^{ - x}}} \right) \\
\Rightarrow y = - {\log _a}\left( {1 - {a^{ - x}}} \right) \\
\]
Therefore,
\[
\Rightarrow 1 - {a^{ - x}} > 0 \\
\Rightarrow {a^{ - x}} < 1 \\
\Rightarrow {a^x} > 1 \\
\Rightarrow x > 0 \\
\]
Hence, the values can be anything greater than 0 to $\infty $.
Therefore, the domain of $y\left( x \right)$ is $\left( {0, + \infty } \right)$. So, our correct option is option (A).
Note:
Note that here we have taken \[1 - {a^{ - x}} > 0\] because, logarithm of a negative number is not possible and hence the value of \[1 - {a^{ - x}}\] must be greater than 0.
Other important point is that we have taken $x > 0$ for ${a^x} > 1$ as we know that ${a^x}$ will be equal to 1 only when $x = 0$, but here ${a^x} > 1$ and so the value of x cannot be 0 and hence, $x > 0$.
Complete step by step solution:
In this question, we are given that y is a function of x and is defined by ${a^{x + y}} = {a^x} + {a^y}$ and we need to find the domain of the function y(x).
Given: ${a^{x + y}} = {a^x} + {a^y}$
First of all, let us see the definition of range.
The domain of a function is the set of all possible input values that produce some output value range.
To find the domain of the given function, we need to simplify it further.
$ \Rightarrow {a^{x + y}} = {a^x} + {a^y}$- - - - - -(1)
Now, we know the property that when two numbers having same base are multiplied, we add their powers. So we can write
${a^{x + y}} = {a^x} \cdot {a^y}$
Therefore, equation (1) becomes
$ \Rightarrow {a^x} \cdot {a^y} = {a^x} + {a^y}$
Now, divide both LHS and RHS with ${a^x} \cdot {a^y}$, we get
\[
\Rightarrow \dfrac{{{a^x} \cdot {a^y}}}{{{a^x} \cdot {a^y}}} = \dfrac{{{a^x} + {a^y}}}{{{a^x} \cdot {a^y}}} \\
\Rightarrow 1 = \dfrac{1}{{{a^y}}} + \dfrac{1}{{{a^x}}} \\
\]
Now, we can write the inverse of any term as $\dfrac{1}{x} = {x^{ - 1}}$. Therefore,
\[
\Rightarrow 1 = {a^{ - x}} + {a^{ - y}} \\
\Rightarrow {a^{ - y}} = 1 - {a^{ - x}} \\
\]
Now, taking log with base a on both sides, we get
\[
\Rightarrow {\log _a}{a^{ - y}} = {\log _a}\left( {1 - {a^{ - x}}} \right) \\
\Rightarrow - y{\log _a}a = {\log _a}\left( {1 - {a^{ - x}}} \right) \\
\]
Now, we know that value of ${\log _a}a = 1$. Therefore, we get
\[
\Rightarrow - y = {\log _a}\left( {1 - {a^{ - x}}} \right) \\
\Rightarrow y = - {\log _a}\left( {1 - {a^{ - x}}} \right) \\
\]
Therefore,
\[
\Rightarrow 1 - {a^{ - x}} > 0 \\
\Rightarrow {a^{ - x}} < 1 \\
\Rightarrow {a^x} > 1 \\
\Rightarrow x > 0 \\
\]
Hence, the values can be anything greater than 0 to $\infty $.
Therefore, the domain of $y\left( x \right)$ is $\left( {0, + \infty } \right)$. So, our correct option is option (A).
Note:
Note that here we have taken \[1 - {a^{ - x}} > 0\] because, logarithm of a negative number is not possible and hence the value of \[1 - {a^{ - x}}\] must be greater than 0.
Other important point is that we have taken $x > 0$ for ${a^x} > 1$ as we know that ${a^x}$ will be equal to 1 only when $x = 0$, but here ${a^x} > 1$ and so the value of x cannot be 0 and hence, $x > 0$.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)