4.05 Inverse functions (Enrichment)
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 f^{-1}(x):
f \left( x \right) = 8 x - 9
f \left( x \right) = x^{2} defined over \left[0, \infty\right)
f \left( x \right) = \sqrt{16 - x^{2}}
State whether the following functions have an inverse function:
f(x) = \dfrac{7}{x}
f(x) = \sqrt{7 - x^{2}}
f(x) = \sqrt{x} + 7
f(x) = 1 - 7 x
f(x) = 7^{x}
f(x) = 4 x^{3} + 8
f(x) = \sqrt{4 + x}
f(x) = - \dfrac{1}{7} x^{2}
State whether the following functions have an inverse function:
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 the given interval.
Find the inverse, f ^{-1}.
Sketch the function f ^{-1}.
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?
The function d\left(t\right) = 120 - 4.9 t^{2} can be used to find the distance, d, that an object dropped from a height of 120 metres has fallen after t seconds.
Is the function d \left(t\right) one-to-one?
Find the inverse function, t \left(d\right).
Prove that d \left( t \left(d\right)\right) = d.
Prove that t \left( d \left(t\right)\right) = t.
Are d and t inverse functions?
How long it will take an object to fall 41.6 metres when dropped from a height of 120 metres?
Consider the function f \left( x \right) = \left(x - 2\right)^{2} - 6. Write the function as two one-to-one functions that have the same rule as f \left( x \right), but are defined over specific intervals.
Consider the graph of y = g \left( x \right) shown:
Complete the piecewise function h(x) below to split the graph of y = g \left( x \right) into three one-to-one functions that have the same rule as g \left( x \right):
h(x) = \begin{cases} g(x), & x\leq ⬚ \\ g(x), & ⬚ \lt x \lt ⬚ \\ g(x), & x \geq ⬚ \end{cases}