Question

If the function f(x) = mx + b has an inverse function, which statement must be true?

(a) m not-equals 0
(b) m = 0
(c) b not-equals 0
(d) b = 0

Answer 1

For the function f(x) = mx + b to have an inverse, the slope m must be non-zero to ensure it's a one-to-one function, making the correct answer (a) m ≠ 0.

Step-by-step explanation:

For the function f(x) = mx + b to have an inverse function, it must be a one-to-one function. That means for every y-value there is a unique x-value and vice versa. In simpler terms, each input should produce a unique output so that the function can be reversed. Therefore, the slope m must be non-zero to ensure that the function is not a horizontal line, which would mean multiple x-values for a single y-value and thus not invertible. Hence, the correct statement must be (a) m ≠ 0. The value of b, which is the y-intercept, does not affect the invertibility of the function. The y-intercept b tells us where the line crosses the y-axis, but does not influence the function's one-to-one nature.