Question

How do you find the domain and range of \[f(x) = {(x + 5)^2} + 8\] ?

Answer 1

Hint: The domain of a function is the complete step of possible values of the independent variable. That is the domain is the set of all possible ‘x’ values which will make the function ‘work’ and will give the output of ‘y’ as a real number. The range of a function is the complete set of all possible resulting values of the dependent variable, after we have substituted the domain.

Complete step-by-step answer:
Given, \[f(x) = {(x + 5)^2} + 8\] .
The domain of the expression is all real numbers except where the expression is undefined means There are no radicals or fractions involved. In this case, there is no real number that makes the expression undefined.
Interval Notation:
 \[\left( { - \infty ,\infty } \right)\]
We can write this in set builder form,
The domain is \[\{ x \in R: - \infty \leqslant x \leqslant + \infty \} \]
The given equation represent the equation of parabola, we have to Locate the vertex of the parabola by using the general vertex form, \[y = a{\left( {x - h} \right)^2} + k\] , to determine the values of \[a\] , \[h\] , and \[k\] .
The given equation as
 \[f(x) = {(x + 5)^2} + 8\]
Where,
 \[a = 1\]
 \[h = - 5\]
 \[k = 8\]
Since the value of \[a\] is negative, the parabola opens down.
The vertex of parabola \[\left( {h,k} \right) = \left( { - 5,8} \right)\]
The range of a parabola that opens down starts at its vertex \[\left( { - 5,8} \right)\] and extends to positive infinity.
Interval Notation: \[\left[ {8,\infty } \right)\]
In set builder form is \[\{ f(x) \in R:f(x) \geqslant 8\} \] . This is the required range and the domain is \[\{ x \in R: - \infty \leqslant x \leqslant + \infty \} \]

Note: The domain where the x values ranges and the range where the y values ranges. Since the equation is a quadratic equation we find the value of x and then we determine the value of y for the values of x. We must know the formula \[y = a{\left( {x - h} \right)^2} + k\] to determine the value of and it is applicable only to this problem since the equation represents the equation of parabola.