Find the domain and range of the function f(x) = √(x-1)

Key Ideas You Must Know

1. Square-root requirement: For a real-valued square-root function $\sqrt{A}$, the radicand $A$ must be non-negative ($A \ge 0$).
2. Domain of a function is the set of all input values for which the expression is defined.
3. Range is the set of all output values the function can actually produce.

Applying the Ideas Step-by-Step

1. Set the radicand \ge 0
   $x - 1 \ge 0$
   $x \ge 1$
   So the domain is $[1,\infty)$.
2. Describe the output
   If $x \ge 1$, the smallest value of $x-1$ is 0. The square-root of 0 is 0.
   As $x$ grows without bound, $x-1$ also grows, and $\sqrt{x-1}$ also grows without bound (though more slowly).
   Hence, $f(x)$ can be any non-negative real number. So the range is $[0,\infty)$.

That is the logical chain a student should follow: first secure the input condition, then see what outputs follow from it.

Simply Put 😊

Imagine you have a magic box that first subtracts 1 from any number you give it, and then it takes the square-root of what is left.

1. What numbers can you safely feed the box?
   You can only subtract 1 and then take a square-root if the part inside the square-root is not negative. So the number you give must be 1 or bigger.
2. What numbers can come out of the box?
   A square-root never throws out negative answers (because we only take the principal or positive root). So the result is always 0 or bigger.

That’s it!
• Domain (allowed inputs): all $x \ge 1$
• Range (possible outputs): all $y \ge 0$

Step-by-Step Solution

1. Write the function $f(x) = \sqrt{x-1}$

2. Find the domain Requirement for a real square-root:
   $x - 1 \ge 0$ $\Rightarrow x \ge 1$ Hence,
   $\text{Domain} = [1,\infty)$

3. Find the range For $x \ge 1$, the expression $x - 1$ is $\ge 0$. The square-root of any non-negative number is also non-negative. The smallest value occurs when $x = 1$:
   $f(1) = \sqrt{1-1} = 0$ As $x \to \infty$, $x-1 \to \infty$ and so $f(x)=\sqrt{x-1} \to \infty$ (unbounded). Hence,
   $\text{Range} = [0,\infty)$

Final Answer
• Domain: $[1,\infty)$
• Range: $[0,\infty)$