5.07 Inverse functions
Explain how a graphing utility such as a graphing calculator be used to visually determine if two functions are inverses of each other.
In each graph below two functions and the line y = x are drawn. For each graph, state whether the functions are inverse functions of each other.
For each of the following functions, find an expression for the inverse function y = f^{ - 1 } \left(x\right):
f \left( x \right) = 7 x - 8
f \left( x \right) = \dfrac{4 x}{3} - 5
f \left( x \right) = 6 x^{3}
f \left( x \right) = \sqrt[3]{x - 2}
f \left( x \right) = \dfrac{8}{x - 9} + 5
f \left( x \right) = \left(x - 4\right)^{3} + 3
f \left( x \right) = \dfrac{8}{5 x + 3}
f \left( x \right) = \dfrac{7}{\left(x + 8\right)^{3}} - 3
f \left( x \right) = 4 \left( 9 x - 2\right)^{3} - 7
For each of the following functions, find:
The domain of f^{-1}(x)
The range of f^{-1}(x)
f \left( x \right) = 8 x - 9 defined over \left[ - 4 , 2\right]
f \left( x \right) = x^{2} defined over \left[0, \infty\right)
f \left( x \right) = \sqrt{16 - x^{2}} defined over \left[0, 4\right]
For each of the following functions, find an expression for the inverse function \\ y = f^{ - 1 } \left(x\right):
f \left( x \right) = - \dfrac{4}{x}
f \left( x \right) = \sqrt[3]{x} + 5
Consider the functions f \left( x \right) = \dfrac{1}{x} + 3 and g \left( x \right) = \dfrac{1}{x - 3}.
Sketch the graph of f \left( x \right).
Sketch the graph of g \left( x \right) on the same set of axes.
Are f \left( x \right) and g \left( x \right) inverses?
Consider the graph of the function f \left( x \right) over the line y = x:
Sketch the graph of f^{ - 1 } \left(x\right).
Consider the graphs of f \left( x \right) and g \left( x \right):
State the equation of f \left( x \right).
State the equation of g \left( x \right).
Evaluate f \left( g \left( x \right) \right).
Evaluate g \left( f \left( x \right) \right).
State whether the following statements are correct:
g \left( x \right) is an inverse of f \left( x \right).
f \left( g \left( x \right) \right) has gradient - 2.
f \left( x \right) is an inverse of g \left( x \right).
g \left( f \left( x \right) \right) has gradient 1.
Consider the graph of y = \dfrac{2}{x} over the line y = x:
Sketch the graph of the inverse of \\y = \dfrac{2}{x}.
Compare the inverse graph to the original graph.
Sketch the graph of the inverse of the following functions:
For each of the following functions:
Sketch the function f \left( x \right) over its domain.
Find the inverse, f ^{-1}.
State the domain of f ^{-1}.
State the range of f ^{-1}.
Sketch the function f ^{-1} over its domain.
f \left( x \right) = x + 6 defined over the interval \left[0, \infty\right).
f \left( x \right) = 7 - x defined over the interval \left[2, 9\right].
f \left( x \right) = \left(x - 6\right)^{2} - 2 defined over the interval \left[6, \infty\right).
f \left( x \right) = \sqrt{4 - x} defined over the interval \left[0, 4\right).
f \left( x \right) = \left(x + 2\right)^{2} + 3 defined over the interval \left[0, \infty\right).
Consider the functions f \left( x \right) = x^{2} - 5 and g \left( x \right) = \sqrt{x + 5}, for x \geq 0. The function y is defined as y = g \left( f \left( x \right) \right), for x \geq 0.
State the equation for y.
Graph the functions f \left( x \right), g \left( x \right) and y on the same set of axes.
What do you notice about the graph of y in relation to the graphs of f \left( x \right) and g \left( x \right)?
The function t = \sqrt{\dfrac{d}{4.9}} can be used to find the number of seconds it takes for an object in Earth's atmosphere to fall d metres.
State the function for d in terms of t.
Find the distance a skydiver has fallen 5 seconds after jumping out of a plane.
The following formula can be used to convert Fahrenheit temperatures x to Celsius temperatures T \left( x \right):
T \left( x \right) = \dfrac{5}{9} \left(x - 32\right)
Find T \left( - 13 \right).
Find T \left( 86 \right).
Find T^{ - 1 } \left(x\right).
What can the formula T^{ - 1 } be used for?
A particle is moving along a straight line path. After t seconds, its velocity is given by the equation v = \left( 96 t - 80\right)^{2}.
Solve for the time t at which the particle comes to rest.
Determine the equation for time, t, in terms of velocity, v, that represents the motion of the particle before it has come to rest.
The tax on a new tablet is 7\% of the advertised price, A.
Determine the equation for the total cost T as a function of the advertised price A.
Hence, express the advertised price A as a function of the total cost T.
The function T = 2 \pi \sqrt{\dfrac{l}{9.8}} can be used to find the period T of a simple pendulum of length l metres.
Express the length l as a function of the period T.
Hence, find the length of a pendulum whose period is 1.5 seconds. Round your answer to one decimal place.