Question

Find an expression for the function whose graph consists of the line segment from.
the point (-2,2) to the point (-1,0) together with the top half of the circle with
centre at the origin and radius 1.​

Answer 1

This is an example of a "piecewise defined function."

On the interval [−2,−1] the function is defined by y=−2x−2 whereas on the interval [−1,1] it is defined by the y=1−x2−−−−−√. Accordingly the function can be expressed in the form

f(x)={−2x−2 for −2≤x<−11−x2−−−−−√ for −1≤x≤1

Another way to express the function is using the unit step function

U(x)={0 for x<01 for x≥0

Then

f(x)=−−2(x+1)U(x+2)+[1−x2−−−−−√+2(x+1)]U(x+1)1−x2−−−−−√U(x−1)

However, this assumes it is defined everywhere on R and equal to 0 outside the given domain.

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Answer 2

$\huge\star\mathfrak\blue{{Answer:-}}$

On the interval [−2,−1][−2,−1] the function is defined by y=−2x−2y=−2x−2 whereas on the interval [−1,1][−1,1] it is defined by the y=1−x2−−−−−√y=1−x2. Accordingly the function can be expressed in the form

f(x)={−2x−2 for −2≤x<−11−x2−−−−−√ for −1≤x≤1

f(x)={−2x−2 for −2≤x<−11−x2 for −1≤x≤1

Another way to express the function is using the unit step functionunit step function

U(x)={0 for x<01 for x≥0

U(x)={0 for x<01 for x≥0

Then

f(x)=−−2(x+1)U(x+2)+[1−x2−−−−−√+2(x+1)]U(x+1)1−x2−−−−−√U(x−1)

f(x)=−2(x+1)U(x+2)+[1−x2+2(x+1)]U(x+1)−1−x2U(x−1)