Question

What is the domain of the function f(x) = square root 3x + 1 for the range 4 <= f(x) <= 5?

Answer 1

`f(x)=\sqrt[]{3x+1}`

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.

`4\leq f(x)\leq5`

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

`\begin{gathered} 4\leq\sqrt[]{3x+1} \\ 4^2\leq(\sqrt[]{3x+1})^2 \\ 16\leq3x+1 \\ 16-1\leq3x \\ 15\leq3x \\ (15)/(3)\leq x \\ \\ 5\leq x \end{gathered}`

When f(x)≤5

`\begin{gathered} 5\ge\sqrt[]{3x+1} \\ 5^2\ge(\sqrt[]{3x+1})^2 \\ 25\ge3x+1 \\ 25-1\ge3x \\ 24\ge3x \\ (24)/(3)\ge x \\ \\ 8\ge x \end{gathered}`

Then. the domain in that interval of range is:

`5\leq x\leq8`