Question

The domain of definition of the function $y\left( x \right)$ given by equation ${{2}^{x}}+{{2}^{y}}=2$ is
A. $0$ $<$x$\le 1$
B. $0$ $\le x$ $\le 1$
C. $-\infty$ $<$x$\le 0$
D. $-\infty$ $<$x$<$ $1$

Answer 1

Hint: In this question we have to find the domain of the given function but first the rules to find the domain must be memorized. Now functions in exponential form must be converted into simple linear variables form by taking logarithm, but first take all other terms on the other side of the equal to sign if they are not terms containing $y$ variable, and we get $y={{\log }_{2}}\left( 2-{{2}^{x}} \right)$ and now put $y=f\left( x \right)$ now the function becomes $f\left( x \right)={{\log }_{2}}\left( 2-{{2}^{x}} \right)$ and now we can find the domain of the function but one thing is to be kept in mind which is that the logarithm function has it's input or domain is all positive real numbers. So put $2-{{2}^{x}}>0$ and then we can evaluate the values of $x$ which is the domain of the original function.

Complete step-by-step answer:
In this question we have to find the domain of the given function but first we need to understand the meaning of the domain of the function which means as inputs of the function at which the function is well defined and does not give an indeterminate form of output or range. Each value of domain only gives one value from the range of the function. Every value from the interval of domain should correspond to a single value from range.
Now before proceeding with the question we must learn all the rules of evaluating the domain of a function which are mentioned below:
1. In square root function there should not be a negative term.
2. The denominator of rational functions should not become zero.
3. The input of logarithmic functions should be positive and the base should also be positive but not equal to 1.
The main point to remember is that the function should be well defined in the interval of it’s domain.
The third point is the one which will be used in this question by us.
Now we need to represent this equation as a proper function and for that we have to take logarithm on both sides of the exponential equation, because the inverse of exponential function is the inverse of logarithm function, but first we need to separate $y$ terms on one side of the equation after which we get,
${{2}^{x}}+{{2}^{y}}=2$
${{2}^{y}}=2-{{2}^{x}}$
Now taking logarithm base 2 on both sides of the equation, we get,
${{\log }_{2}}{{2}^{y}}={{\log }_{2}}\left( 2-{{2}^{x}} \right)$
Now applying the property of logarithm function which is ${{\log }_{a}}{{x}^{n}}=n{{\log }_{a}}x$, we get,
$y{{\log }_{2}}2={{\log }_{2}}\left( 2-{{2}^{x}} \right)$
Now applying the second property of logarithm function which is ${{\log }_{a}}a=1$, we get,
$y={{\log }_{2}}\left( 2-{{2}^{x}} \right)$
Now we have simplified our function so we can calculate the domain of the function by applying the third point in the above-mentioned rules of evaluating the domain of a function containing a logarithmic term.
Making the logarithmic function term greater than zero, we get,
$2-{{2}^{x}}>0$
$2>{{2}^{x}}$
Now if $x$ is equal to 1 then we will get 2 but we need the R.H.S. less than 2 and if we put $x$ greater than 1 then it will result in number greater than 2, so there is only one option left which is having $x$ less than 1 and hence we will get numbers less than 2,
Therefore the inequality becomes,
$x<1$, after taking the logarithm of base 2 on both sides we end up with this inequality.
Hence the interval of domain of the function is $\left( -\infty ,1 \right)$ and $x$ belongs to this interval also.
The correct answer is,
Option. D. $-\infty$ $<$x$<$ $1$.

Note:One should remember all the properties of logarithm function like ${{\log }_{a}}x=\dfrac{{{\log }_{c}}x}{{{\log }_{c}}a}$, ${{\log }_{{{a}^{n}}}}x=\dfrac{1}{n}{{\log }_{a}}x$, ${{a}^{{{\log }_{a}}x}}=x$ and many more. All the rules of domain calculations for each and every function as mentioned above shall also be well understood by looking and constructing graphs of the functions like modulus function, greatest integer function, logarithm function, exponential function parabola function and many more. Remember the interval of domain of the function should have each point applicable as input of the function and every point should correspond to only a single value, whether same or different for each point it does not matter, and not multiple values of the range of function. When calculating the range of a function we need to form it’s inverse and then calculate the domain of the inverse of the original function which will act as a range of the original function.