Question

Find the domain of the piecewise function and evaluate it for the given values. Then use the drop down menu to select the correct symbols to indicate your answer in interval notation. If a number is not an integer then round it to the nearest hundredth. To indicate positive infinifty ( \infty ) type the three letters "inf". To indicate negative infinity(-\infty ) type "-inf" with no spaces between characters. f(x) = \left\lbrace \begin{array}{cc} x^2-2 & x<2 \\ 4+|x-5| & x\ge 2 \end{array}\right. The domain is:AnswerAnswer,AnswerAnswerf(-1)=Answerf(0)=Answerf(2)=Answerf(4)=Answer

Answer 1

Given

`f(x)=\begin{cases}x^2-2\text{ x<2} \\ 4+|x-5|\text{ x}\ge2\end{cases}`

We want to find the domain, f(-1), f(0), f(2), f(4)

Solution

To find the domain

We can see that

`\begin{gathered} \text{first,} \\ x<2\text{ which is (-inf,2)} \\ we\text{ also have} \\ x\ge2\text{ which is \lbrack{}2,inf)} \\ so\text{ we take the union} \\ (-\text{inf, 2)U\lbrack{}2, inf) = (-inf, inf)} \end{gathered}`

The domain is (-inf, inf)

To find f(-1)

Since -1<2, we use

`\begin{gathered} f(x)=x^2-2 \\ f(-1)=(-1)^2-2=1-2=-1 \end{gathered}`

To find f(0)

Since 0<2, we use

`\begin{gathered} f(x)=x^2-2 \\ f(0)=0^2-2=-2 \end{gathered}`

To find f(2)

Now, since x=2, we use

`\begin{gathered} f(x)=4+|x-5| \\ f(2)=4+|2-5| \\ f(2)=4+|-3| \\ f(2)=4+3 \\ f(2)=7 \end{gathered}`

To find f(4)

Again x=4>2, we still use

`\begin{gathered} f(x)=4+|x-5| \\ f(4)=4+|4-5| \\ f(4)=4+1 \\ f(4)=5 \end{gathered}`