[tex]f(x)=\sqrt[]{3x+1}[/tex]
The domain of a functio f(x) is the set of vlues for which the function is defined (all the values of x for which the function is defined).
Range: the values that f(x) takes.
[tex]4\leq f(x)\leq5[/tex]
Find the values of x when f(x) is greather than or equal to 4 and less than or equal to 5
When f(x)≥4
[tex]\begin{gathered} 4\leq\sqrt[]{3x+1} \\ 4^2\leq(\sqrt[]{3x+1})^2 \\ 16\leq3x+1 \\ 16-1\leq3x \\ 15\leq3x \\ \frac{15}{3}\leq x \\ \\ 5\leq x \end{gathered}[/tex]
When f(x)≤5
[tex]\begin{gathered} 5\ge\sqrt[]{3x+1} \\ 5^2\ge(\sqrt[]{3x+1})^2 \\ 25\ge3x+1 \\ 25-1\ge3x \\ 24\ge3x \\ \frac{24}{3}\ge x \\ \\ 8\ge x \end{gathered}[/tex]
Then. the domain in that interval of range is:
[tex]5\leq x\leq8[/tex]
