R
where g(x) = 2x + 5 for all x in R is
a one-to-one correspondence (that is, one-to-one and onto). 8. First determine if the function
g: RR
where g(x) = 2x + 5 for all x in R is
a one-to-one correspondence (that is, one-to-one and onto).
The function g is one-to-one. Suppose that x1 and x2 are both real numbers such that g(x1) = g(x2). Then 2x1 + 5 = 2x2 + 5, and it follows that x1 = x2.
The function g is onto. Suppose that y is some real number. Then there always exists a real number x, namely x = (y - 5)/2, such that g(x) = y.
To find the inverse function of g, by definition we know that g-1(y) is that unique real number x such that g(x) = y. We know that
| g(x) | = | y |
| 2x + 5 | = | y |
| x | = | y - 5 |
| 2 |
Hence
g-1: R
R
where
g-1(y) = (y - 5)/2.