Question

Find the domain and range of $f$.

$f\left(x\right)=\left\{\begin{array}{c}1,\text{ifxisrational}\\ 5,\text{ifxisirrational}\end{array}\right\}$

Short answer

Domain of f is $\left(-\infty ,\infty \right)$

Range of f is$\left\{1,5\right\}$

Step-by-step solution

Step 1. Use the definition of functions.

A function is also defined as, given a set of inputs X and a set of possible outputs Y as a set of ordered pairs (x,y). We can write the statement that f is a function from X to Y using the function notationf:$X\to Y$.

This means that if the object x is in the set of inputs (called the domain) then a function f will map the to exactly one object f(x) in the set of possible outputs (called the range).

Step 2. find the domain.

We are the piecewise function: $f\left(x\right)=\left\{\begin{array}{c}1,\text{ifxisrational}\\ 5,\text{ifxisirrational}\end{array}\right\}$, since function is defined for both rational and irrational values of x, thus function is defined for all possible values of x hence domain is $\left(-\infty ,\infty \right)$.

Step 3. find the range.

We are the piecewise function: $f\left(x\right)=\left\{\begin{array}{c}1,\text{ifxisrational}\\ 5,\text{ifxisirrational}\end{array}\right\}$, since function can attain only two values either 1 or 5 hence its range is $\left\{1,5\right\}$.