What happens if we graph both f and f -1
on the same set of axes, using the x-axis for
the input to both f and f -1 ?
f -1 is the inverse of the function f. Although it may seem as if it’s being raised by the -1 power
in reality it isn’t. For any one-on-one function f(x), the inverse is a function denoted by f -1(x)
such that f -1(f(x)) = x for all x in the domain of f: this also implies that f (f -1(x)) = x for all x
in the domain of f -1. We can take the output and find the input with the idea of the inverse
function. A typical function gives us the output given any input and the inverse function gives
us the input given any output. A function and its inverse are symmetric about the line y=x. In
essence when you reflect the function f(x) about the line y=x, the graph produce in its inverse
somewhat like a mirror effect.
Type y = x3 {-2 < x < 2}, y = x 1/3 {–2 < x < 2}, and y = x {–2 < x < 2}, and describe the
relationship between the three curves. Then post your own example discussing the
difficulty of graph both f and f -1 on the same set of axes.
When you reflect the graph y=x3 about the line y=x you get y=x1/3. y=x1/3 is the inverse of
y=x3.
Fig. 1.1: Showing a graph of y = x3; y = x1/3; y = x
Fig. 1.2 : Showing a graph of f and f -1
on the same set of axes, using the x-axis for the
input to both f and f -1
y = x2, x≥ 0
x = y2
√ x =√ y 2
√x=y
f -1(x) = x1/2, x≥0
on the same set of axes, using the x-axis for
the input to both f and f -1 ?
f -1 is the inverse of the function f. Although it may seem as if it’s being raised by the -1 power
in reality it isn’t. For any one-on-one function f(x), the inverse is a function denoted by f -1(x)
such that f -1(f(x)) = x for all x in the domain of f: this also implies that f (f -1(x)) = x for all x
in the domain of f -1. We can take the output and find the input with the idea of the inverse
function. A typical function gives us the output given any input and the inverse function gives
us the input given any output. A function and its inverse are symmetric about the line y=x. In
essence when you reflect the function f(x) about the line y=x, the graph produce in its inverse
somewhat like a mirror effect.
Type y = x3 {-2 < x < 2}, y = x 1/3 {–2 < x < 2}, and y = x {–2 < x < 2}, and describe the
relationship between the three curves. Then post your own example discussing the
difficulty of graph both f and f -1 on the same set of axes.
When you reflect the graph y=x3 about the line y=x you get y=x1/3. y=x1/3 is the inverse of
y=x3.
Fig. 1.1: Showing a graph of y = x3; y = x1/3; y = x
Fig. 1.2 : Showing a graph of f and f -1
on the same set of axes, using the x-axis for the
input to both f and f -1
y = x2, x≥ 0
x = y2
√ x =√ y 2
√x=y
f -1(x) = x1/2, x≥0
