Which answer is true

Question

Suppose that for a function f, the equation f(x)=0 has not real-number solutions.

 

Which of the following must be true?

 

A. f(x)>0 for all x in the domain of f

B. f has no x-intercepts

C. f has no y-intercepts

D. f is an invertible function

Answer 1

The function could satisfy f(x) < 0 for all x in the domain. This eliminates A.

 

f(x) = x² + 1 has no real roots but has y-intercept at x=0 , y=1. This eliminates C.

 

The same function has no inverse since it fails the horizontal line test, for example, x² + 1 = 2 has two solutions ±1. This eliminates D.

 

The answer must therefore be choice B.

 

Another way to solve this problem is to realize that an x-intercept is a point on the graph where f(x) = 0. So if a function had an x-intercept, it would also have a real root. This once again yields choice B as the answer.

Answer 2

When we set f(x)=0, we are actually finding the x-intercepts.

 

 

If f(x) has no real-number solutions, then f(x) never touches nor crosses the x-axis.    Therefore, f(x) has no x-intercepts.