let g be a function such that g(-1)=0 and g(2)=5. which of the following conditions guarantees that there is an x, -1<x<2, for which g(x)=3?

Question

let g be a function such that g(-1)=0 and g(2)=5. which of the following conditions guarantees that there is an x, -1<x<2, for which g(x)=3?

a) g is defined for all x in (-1,2)

b) g is continuous for all x in [-1,2]

c) g is increasing on [-1,2]

d) there exists an x in (-1,2) such that g(x)=6

please show works

Answer 1

The answer is given by the intermediate value theorem which states that if we have a continuous function f defined on a closed interval [a,b] and c is any number between min{f(a),f(b)} and max{f(a),f(b)} then there exists an x in (a,b) such that f(x)=c. So you need your function g to be continuous on the interval [-1,2] in order to guarantee the existence of an x such that g(x)=3.