Explain the range of f (x)=sqrt(x-1) by equating it to y

To find the range of the function \( f(x) = \sqrt{x - 1} \), we can start by setting it equal to \( y \). This gives us the equation:

Setting Up the Equation

We rewrite the function as:

y = √(x - 1)

Isolating x

Next, we can square both sides to eliminate the square root:

y² = x - 1

Now, we can solve for \( x \):

x = y² + 1

Determining the Values of y

Since \( y \) represents the output of the square root function, it must be non-negative. This means:

• y ≥ 0

Finding the Range

As \( x \) increases from 1 (the minimum value for which the function is defined), \( y \) starts from 0 and can increase indefinitely. Therefore, the range of \( f(x) \) is:

Range: [0, ∞)

Summary

The function \( f(x) = \sqrt{x - 1} \) has a range that starts at 0 and goes to positive infinity, indicating that it can take any non-negative value.