Domain and Range

Worked example

1. 1

   $g(x)=\frac{4x}{x+9}$ — Look at the bottom of the fraction. Only the bottom can cause trouble.

2. 2

   $x+9=0$ — A fraction with 0 on the bottom has no value, so find the x that makes the bottom 0.

3. 3

   $x=-9$ — Take 9 from both sides. Putting in x = −9 would put 0 on the bottom.

4. 4

   $x\ne -9$ — Write the answer as x ≠ −9. Every other value of x is allowed.

Worked example

1. 1

   $k(x)=\frac{7}{3-4x}$ — A second one, with a minus in front of the x. Look at the bottom again.

2. 2

   $3-4x=0$ — Set the bottom equal to 0.

3. 3

   $-4x=-3$ — Take 3 from both sides.

4. 4

   $x=\frac{-3}{-4}=\frac{3}{4}$ — Divide both sides by −4. A negative divided by a negative is positive.

5. 5

   $x\ne \frac{3}{4}$ — Leave out x = 3/4, or 0.75.

Worked example

1. 1

   $g(x)=\sqrt{8-2x}$ — Look at what is inside the root.

2. 2

   $8-2x\ge 0$ — You cannot take the square root of a negative, so the inside must be 0 or more.

3. 3

   $8\ge 2x$ — Add 2x to both sides. This avoids dividing by a negative number later.

4. 4

   $4\ge x$ — Divide both sides by 2.

5. 5

   $x\le 4$ — Read it the other way round: x can be 4 or smaller.

6. 6

   $x>4\text{is left out}$ — The question asks which values cannot be used, so give the opposite: anything above 4.

Worked example

1. 1

   $f(x)=(x+2{)}^2$ — A squared bracket.

2. 2

   $(x+2{)}^2\ge 0$ — A square is never negative, so the squared bracket is 0 or more.

3. 3

   $x+2=0\Rightarrow f(-2)=(-2+2{)}^2=0$ — The lowest value is 0. It happens when the bracket is 0, at x = −2.

4. 4

   $f(x)\ge 0$ — There is no top limit, because x can be as large as you like. Write the range with f(x).

Worked example

1. 1

   $f(x)=(x-1{)}^2+5$ — A squared bracket, plus 5.

2. 2

   $(x-1{)}^2\ge 0$ — A square is never negative.

3. 3

   $(x-1{)}^2+5\ge 0+5$ — Add 5 to both sides, so the whole function is 5 or more.

4. 4

   $f(x)\ge 5$ — Write the range with f(x).

Worked example

1. 1

   $h(x)=x^2+4x,x\ge -2$ — Complete the square first, to find the lowest point.

2. 2

   $x^2+4x=(x+2{)}^2-4$ — Half of 4 is 2, so use (x + 2)². That bracket adds an extra 4, so take 4 away.

3. 3

   $h(-2)=(-2+2{)}^2-4=-4$ — The domain starts at x = −2, exactly where the bracket is 0. So −4 is the lowest value.

4. 4

   $h(x)\ge -4$ — The values rise from −4 with no top limit.

Worked example

1. 1

   $p(x)=4-3x,\text{domain }\{-2,0,3\}$ — Put each number of the domain in, one at a time.

2. 2

   $p(-2)=4-3\times (-2)$ — Replace x with −2. Use a bracket to keep the negative sign.

3. 3

   $p(-2)=4+6=10$ — −3 × −2 = +6, because a minus times a minus is a plus.

4. 4

   $p(0)=4-0=4$ — Replace x with 0.

5. 5

   $p(3)=4-9=-5$ — Replace x with 3, then work out 4 − 9.

6. 6

   $\text{range}=\{-5,4,10\}$ — List the outputs, smallest first. These three numbers are the whole range.

Worked example

1. 1

   $h(x)=x^2+4x,x\ge -2$ — The function and its domain.

2. 2

   $\text{range of }{h}^{-1}=\text{domain of }h$ — An inverse swaps inputs and outputs, so the inputs of h become the outputs of the inverse.

3. 3

   $h^{-1}(x)\ge -2$ — The domain of h starts at −2, so every output of the inverse is −2 or more.