Finding the value for the constant a so that the function is continuous

Question

Given the function f(x)= {a-x if x≤𝜋}

{asin(x) if x>𝜋}

a) find the value for the constant a so that the function is continuous

b) show that the function is continuous at x = 𝜋 for that value of a

c) sketch the graph of the function using that value of a

Answer 1

Hi Naeema,

The function given is sometimes called a "piecewise" function since it has two different equations for two different domains.

For part a) of the question, we need to determine a value for "a" that will make the function continuous at the point where the function switches from one equation to the other. That switching point is at x=pi.

To be continuous, the 2 equations in the piecewise function must be equal at x=pi.

So, we need to think about those 2 equations and find where they intersect.

Find where they intersect by setting those 2 equations equal to each other.

Equation 1) f(x)=a-x

Equation 2) f(x)=a*sin(x)

Now,

a-x = a*sin(x)

We can't really solve this equation for a easily, so let's assume a=1 and graph these on your calculator:

y=1-x

y=sin(x)

See that these two functions intersect at (0.511,0.489). We need them to intersect at x=pi, not x=0.511.

So, a=1 is wrong.

Where is the function sin(x) at x=pi? It's (3.14,0). Ok, so we know any coefficient "a" on f(x)=a*sin(x) will only change amplitude, not this point at (3.14,0).

So, the 2 equations need to intersect at this point (3.14,0).

Therefore, "a" must be adjusted so that the first equation (a-x) will intersect at point (3.14,0) too.

This means a=pi. That way, the equations are:

y=pi-x

y=pi*sin(x)

See that these intersect at point (3.14,0). That's the answer to part a).

Part b) You can prove that it is continuous by writing that the limits as x approaches pi from both positive and negative directions are the same and equal to the function when x=pi. In all cases f(pi)=0 and the limits from both directions =0, so therefore it is continuous. You can also show by graphing.

Part c) Go ahead and sketch! Use graphing calculator for guidance. Make sure you graph y=pi-x for the domain -infinity to pi (including pi), and graph y=pi*sin(x) for the domain pi to infinity (not including pi).

Hope this helps! :)