Determine whether the theorem applies to the given situation by determining whet

Question

Determine whether the theorem applies to the given situation by determining whether the hypothesis of the theorem is satisfied. If the theorem applies, draw a specific conclusion. If not. explain how the hypothesis fails to be satisfied. Thin: If a function f is defined for every real number, then its graph has no vertical asymptotes. Given: f(x) =1/x. Thin: If a real number v is positive, then it has a reciprocal. Given: x = 0 Thin: If a real number v is nonnegative, then it has a square root. Given: x = 0 Thin: If an expression can be written as a difference of squares, then it is factorable. Given: x^4 - 16 Thin: If the graph of f is a non-horizontal line, then its slope is nonzero. Given: f(x) = 6

Explanation / Answer

8) The theorem doesn't apply to the given situation, since the function f(x) = 1/x is not defiend for the the value of x equal to zero

Hence the line x=0 will be a vertical asymptote in the graph

9) The theorem is applied, since the x is a real positive number, hence it can't be zero

Let the reciprocal of x be x'

xx' = 1

x' = 1/x

We can keep the x in the denominator since x is not equal to zero

10) The statement is TRUE

Reason:

Since x is positive, hence

y = sqrt(x), will exists because it will give a real solution

11) The statement is TRUE

It can be written as difference of squares which is factorable

x^4 - 16 = (x^2-4)(x^2+4) = (x-2)(x+2)(x^2+4)

12) The statement is FALSE

y = 6

dy/dx = 0, hence the slope of the graph is equal to zero

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