Question

How do you find the domain and range of \[g(x)\, = \,\dfrac{x}{{{x^{^2}} - 16}}\] ?

Answer 1

Hint: In order to determine the domain from the given equation is a function. To find out the domain, look for those values of the independent variable (usually \[x\] ) which we are allowed to use. The range of a function is the complete set of all possible resulting values of the dependent variable ( \[y\] , usually), after we have substituted the domain.

Complete step by step solution:
We are given the range of \[g(x)\, = \,\dfrac{x}{{{x^{^2}} - 16}}\]
To calculate the domain i.e. finding out the value of \[x\] , the denominator cannot be zero as you cannot divide by zero.
 \[ \Rightarrow {x^{^2}} - 16 \ne 0\]
So, the value of x can be:
 \[ \Rightarrow x = \pm \sqrt {16} \]
 \[ \Rightarrow x = \pm 4\]
 \[\therefore \] The domain of g(x) is \[x \in R - \left\{ { - 4,4} \right\}\] .
To calculate the range \[y\] ,
Let \[y = \dfrac{x}{{{x^2} - 16}}\]
Now multiplying the denominator on left hand side, we get
 \[ \Rightarrow y({x^2} - 16) = x\]
On simplifying it in form of quadratic equation we get
 \[ \Rightarrow y{x^2} - x - 16 = 0\]
Now, we can use the formula under the radical, \[{b^2} - 4ac\] , called the discriminant, to determine the number of roots of solutions in the above quadratic equation.
Here \[a = y,b = - 1\] and \[c = - 16y\] so
 \[ \Rightarrow {( - 1)^2} - 4(y)( - 16y)\]
On further simplifying we get
 \[ \Rightarrow 1 + 64{y^{^2}}\]
 \[\therefore y\, \in \,R\] so the range of the given function is \[g(x)\, \in \,R\] .
As a result, the domain is \[x \in R - \left\{ { - 4,4} \right\}\]
So, the correct answer is “ \[x \in R - \left\{ { - 4,4} \right\}\] ”.

Note: When finding the domain, remember:
The denominator (bottom) of a fraction cannot be zero and the number under a square root sign must be positive in this section
When finding the range, remember: Substitute different x-values into the expression for y to see what is happening. Make sure you look for minimum and maximum values of y. Alternatively, range can be found out with the help of graphs.