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Here is a table of all input/output pairs
for a one-to-one function \(f.\)
What is \(f^{-1}(3)?\)
input \(x\) -3 -2 -1 0 1 2 3 output \(f(x)\) -2 1 4 3 -3 -6 2 -
These functions are one-to-one on their implied domains.
For each, write down a formula for its inverse.
\(\displaystyle g(x) = \frac{x}{11}-42\)\(\displaystyle h(x) = \frac{2}{3}x^{1.416}\)\(\displaystyle k(x) = \sqrt[3]{3+x^7}-\frac{1}{4}\)
- The function with formula \(f(t) = \frac{5}{9}(t-32)\) takes a temperature \(t\) measured in Fahrenheit (°F) and returns that temperature measured in Celsius (°C). The function with formula \(g(t) = t+273.15\) takes a temperature \(t\) measured in °C and returns that temperature measured in Kelvins (K).
- Write down a formula for the function that take a temperature measured in °F and returns that temperature measured in Kelvins.
- Write down a formula for the function that take a temperature measured in °C and returns that temperature measured in °F.
- Write down a formula for the function that take a temperature measured in Kelvins. and returns that temperature measured in °F.
Challenge
Considering the domain of all real numbers, the function \(x^3\) is one-to-one, but \(x^2\) is not. What about if the exponent is a fraction? or a decimal number? The function \(h(x) = \frac{2}{3}x^{1.416}\) above was declared to be one-to-one, but is it? How do we know?
Big question: for what numbers \(b\) is the power function \(f(x) = x^b\) one-to-one?
- The function with formula \(f(t) = \frac{5}{9}(t-32)\) takes a temperature \(t\) measured in Fahrenheit (°F) and returns that temperature measured in Celsius (°C). The function with formula \(g(t) = t+273.15\) takes a temperature \(t\) measured in °C and returns that temperature measured in Kelvins (K).