Solution

For graphing:

• Use a small table of convenient $x$ values.
• Note transformations (shifts, reflections, stretches).
• State the domain restriction if a square root is involved.

(i) $y=\sqrt{3x}$

Domain: $3x\ge 0\Rightarrow x\ge 0$.

Choose $3x$ as perfect squares: $3x=0,1,4,9,16$.

(ii) $y=\sqrt{x}+5$

Domain: $x\ge 0$.

This is $y=\sqrt{x}$ shifted up by 5.

(iii) $y=-\dfrac{1}{2}\sqrt{x}$

Domain: $x\ge 0$.

Reflection in $x$-axis and vertical shrink by factor $\frac{1}{2}$.

(iv) $y=-\sqrt{x+1}+2$

Domain: $x+1\ge 0\Rightarrow x\ge -1$.

Choose $x+1=0,1,4,9,16$.

This is $y=\sqrt{x}$ shifted left by 1, reflected, then shifted up by 2.

(v) $y=\sqrt[3]{2x+1}$

Cube root is defined for all real numbers.

Pick $2x+1=-8,-1,0,1,8$.

(vi) $y=2\sqrt[3]{x}-3$

Pick $x=-8,-1,0,1,8$.

Stretch by 2 and shift down by 3.

(vii) $y=\sqrt[3]{x^2+x-2}$

Cube root is defined for all real numbers, so domain is all real numbers.

Compute a few points: