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# showing continuity of functions

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I have the following theorem "If the function g is continuous everywhere and the function f is continuous everywhere, then the composition of f o is continuous everywhere."

How can I directly apply this theorem to show that the equations

1) sin(x^3+7x+1)

2) |sinx|

3) cos^3 (x+1)

are continuous everywhere?

Jun 9, 2022

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Note that f o g    means we are putting  g  into f

1)    Let    f  =   sin x    .......  a sinusoidal  function is  continuous everywhere

Let    g =  x^3 + 7x + 1......this is a polynomial....so it is continuous everywhere

So   f  o  g  =       sin (x^3 + 7x + 1)    is continuous

2)   Let   f =    lx l  .......the basic absolute value function is  contimuous everywhere

Let  g =   sin x    (continuous  everywhere )

So    f o  g  =    l sin x  l    is continuous

3)    As before

f  = cos ^3  x          sinusoidal function { continuous  everywhere }

g =  x +1                linear function  {continuous everywhere}

So

f o g  =    cos^3 ( x + 1)      is continuous   Jun 9, 2022