For the following exercises, find the domain of each function using interval notation.

Expert intro: To find a domain, look for restrictions such as square roots and polynomial expressions. Polynomials are defined for all real numbers, while square roots require the radicand to be nonnegative.

Detailed walkthrough

Find the domain of each function in interval notation.

6. $f(x)=-2x(x-1)(x-2)$

This is a polynomial, so it is defined for all real numbers.

Domain: $(-\infty,\infty)$

7. $f(x)=5-2x^2$

This is also a polynomial.

Domain: $(-\infty,\infty)$

8. $f(x)=3\sqrt{x-2}$

The expression under the square root must be nonnegative:

$$x-2\ge 0 \Rightarrow x\ge 2$$

Domain: $[2,\infty)$

9. $f(x)=3-\sqrt{6-2x}$

Require

$$6-2x\ge 0 \Rightarrow -2x\ge -6 \Rightarrow x\le 3$$

Domain: $(-\infty,3]$

10. $f(x)=\sqrt{4-3x}$

Require

$$4-3x\ge 0 \Rightarrow -3x\ge -4 \Rightarrow x\le \frac43$$

Domain: $(-\infty,\frac43]$

11. $f(x)=\sqrt{x^2+4}$

Since $x^2\ge 0$ for all real $x$, we have

$$x^2+4\ge 4>0$$

Domain: $(-\infty,\infty)$

💡 Pitfall guide

For square roots, do not solve by setting the inside equal to zero only. You need the inside to be greater than or equal to zero. Also, remember that polynomials have no domain restrictions.

🔄 Real-world variant

If a function had a denominator, such as $\frac{1}{\sqrt{x-2}}$, the domain would change because the radicand would need to be strictly positive, not just nonnegative.

🔍 Related terms

domain, interval notation, radical function