Question

3A. Explain in your own words how to find the domain of a function if you know its equation.B. Find the domain of f (x) =V9-7x6x2+13x-15Be sure to show relevant work.

Answer 1

Part A

The domain of a function represents the possible values of x which make the function exist.

hence when finding the domain of a function, the denominator cannnot be zero because it will become undefine and the square root sign must not be negative

Par B

the function given is

`f(x)=\frac{\sqrt[]{9-7x}}{6x^2+13x-15}`

To find the domain, the denominator must not be zero

`\begin{gathered} 6x^2+13x-15=0 \\ 6x^2+18x-5x-15=0 \end{gathered}`
`\begin{gathered} 6x(x+3)-5(x+3)=0 \\ (6x_{}-5)(x+3)=0 \end{gathered}`
`\begin{gathered} \text{hence} \\ x=(5)/(6)\text{ or x=-3} \end{gathered}`

Hence 5/6 and -3 make the function undefine

Also, The function is undefined for all the values of x where

`\begin{gathered} 9-7x<0 \\ 9<7x \\ (9)/(7)(9)/(7) \end{gathered}`

The function is undefined for all the values of x>9/7

Hence the domain of the function is given by

`(-\infty,-3)\cup(-3,(5)/(6))\cup((5)/(6),(9)/(7))`

Therefore the domain of the function is (-∞ ,-3) U (-3,5/6) U (5/6,9/7)