5.07 Inverses and their graphs
Find the inverse of the following functions:
y = - 2 \left(x + 2\right)
y = \dfrac{4 x - 3}{5}
y = 6 - x^{3}
y = x^{5} + 2
y = \dfrac{1}{x - 8}
y = \dfrac{7 x + 6}{x - 7}
f \left(x\right) = 5 x
y = - 8 x + 6
y = \dfrac{2}{9} x
y = \dfrac{1}{8} x + 7
y = \dfrac{1}{7} x - 1
For each of the following pair of functions:
Evaluate f\left(g\left(x\right)\right).
Evaluate g\left(f\left(x\right)\right).
Are f \left( x \right) and g \left( x \right) inverses?
f \left( x \right) = - \dfrac{8}{9} x and g \left( x \right) = - \dfrac{9}{8} x
f \left( x \right) = 4 x + 16 and g \left( x \right) = \dfrac{1}{4} x - 4
f \left( x \right) = 3 x - 12 and g \left( x \right) = \dfrac{1}{3} x - 4
f \left( x \right) = x^{3} - 5 and g \left( x \right) = \sqrt[3]{x + 5}
State whether the following functions have an inverse function:
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.
Sketch the graph of the inverse of the following functions:
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 graph of y = \dfrac{1}{x} over the line y = x:
Sketch the graph of the inverse of \\y = \dfrac{1}{x}.
Compare the inverse graph to the original graph.
Consider the functions f \left( x \right) = \dfrac{1}{x} + 5 and g \left( x \right) = \dfrac{1}{x - 5}.
Sketch the graph of f \left( x \right).
Sketch the graph of g \left( x \right) on the same number plane.
Are f \left( x \right) and g \left( x \right) inverses?
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 -\dfrac{2}{3}.
f \left( x \right) is an inverse of g \left( x \right).
g \left( f \left( x \right) \right) has gradient 1.
A ski resort has introduced a new ski pass system whereby everyone needs to purchase an electronic card for a one time fee of \$8. They can then put credit on the card for each day's lift access. Lift access costs \$80 per day.
Let C \left(x\right) = 8 + 80 x represent the cost of purchasing x days worth of lift access on the card.
Define y, the inverse of C \left(x\right) = 8 + 80 x.
What does the inverse function y = C^{ - 1 } \left(x\right) represent?
Determine the number of days' lift access, y, that can be purchased with \$728.
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, where t \geq 0.
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?