Chapter 2, Problem 7RE

In Problems 6-11, find the domain of each function.

$f\left(x\right)\text{ = }\sqrt{2-x}$

To determine

To find: The domain of the provided function.

Answer to Problem 7RE

Solution:

The domain of $f$ is $\left\{x\right|x\ge 2\}$, or the interval $\left[2,\infty \right)$.

Explanation of Solution

Given:

It is provided in the problem that the function is $f\left(x\right)=\sqrt{2-x}$.

Formula used:

Rules used to find the domain of a function defined by an equation:

1. Start with the domain as the set of all numbers.

2. If the equation has a denominator, exclude any numbers that give a zero denominator.

3. If the equation has a radical of even index, exclude any numbers that cause the expression inside the radical (the radicand) to be negative.

The function $f$ says to take the square root of $2-x$. But only nonnegative numbers have real squre roots, so the expression under the squre root (the radicand) must be nonnegative (greater than or equal to zero). This requires that

$2-x\ge 0$

$x\ge 2$

The domain of $f$ is $\left\{x\right|x\ge 2\}$, or the interval $\left[2,\infty \right)$.