Question

Determine the domains of the following functions. Write your answer in interval nota-tion. Exercise: b

Answer 1

#(a)

The given function is

`f(x)=3-\sqrt[]{10-2x}`

Since there is no square root for a negative value, then put (10 - 2x) greater than or equal zero to find all values of x can make the function f(x) defined

`10-2x\ge0`

Add 2x to both sides

`\begin{gathered} 10-2x+2x=0+2x \\ 10\ge2x \\ 2x\leq10 \end{gathered}`

Divide both sides by 2

`\begin{gathered} (2x)/(2)\leq(10)/(2) \\ x\leq5 \end{gathered}`

The domain is all values of x less than or equal to 5

`D=(-\infty,5\rbrack`

#b)

The given function is

`g(x)=(17x-1)/(x^2-4x+3)`

Equate the denominator by 0, then factorize it into 2 factors to find the values of x which make the denominator equal to 0, then exclude these values from the real number

`\begin{gathered} x^2-4x+3=0 \\ (x-3)(x-1)=0 \end{gathered}`

Equate each factor by 0

`\begin{gathered} x-3=0 \\ x-3+3=0+3 \\ x=3 \end{gathered}`
`\begin{gathered} x-1=0 \\ x-1+1=0+1 \\ x=1 \end{gathered}`

The domain of g(x) is all real numbers except 1 and 3

`D=(-\infty,1)\cup(1,3)\cup(3,\infty)`