Which of the following functions has a range of all real numbers?

Question

y=-3x^2+5

y=4x^2-1

y=-7x+2

y=6(2^X)

Answer 1

Some facts to use:

 

1. A quadratic ax²+bx+c has a vertex when x = -b/2a, and this vertex is an absolute minimum or maximum. Can you put some of these functions into this form?

 

2. A positive number raised to any value positive, zero, or negative, is always positive.

 

Now, to have range all real numbers the function must reach every height on the y axis. It must have a solution to y₀=f(x) for every y₀. That is why the first fact is useful: because a quadratic reaches an absolute minimum or maximum, so a point below that or above that, respectively, in the y direction, it never reaches. Likewise, 2. is useful because an exponential with a positive base never reaches 0 or a negative number.

 

Finally, on the other hand we suppose we are trying to reach a given value y₀. Then we can set y₀ = -7x+2, and we can solve this no matter what y₀ is to find the value of x such that we get the given value of y₀. Visually, this is because this is a line with non-zero slope, so it transverses the entire range of y values. It goes "all the way up" in one direction, and "all the way down" in the other.