Inverse Functions

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In this activity, you will discover the relationship between functions and inverse functions.

Write the operation of adding 3 as a function of $x$ :

$y= \answer {x+3}$ .

What operation undoes adding 3? Write this operation as a function of $x$ .

$y = \answer {x-3}$

Write the operation of subtracting 8 as a function of $x$ :

$y= \answer {x-8}$ .

What operation undoes subtracting 8? Write this operation as a function of $x$ .

$y=\answer {x+8}$

Write the operation of multiplying by $-2$ as a function of $x$ :

$y= \answer {-2x}$ .

What operation undoes multiplying by $-2$ ? Write this operation as a function of $x$ .

$y=\answer {\frac {x}{-2}}$

Write the operation of dividing by 6 as a function of $x$ :

$y= \answer {\frac {x}{6}}$ .

What operation undoes dividing by 6? Write this operation as a function of $x$ .

$y=\answer {6x}$

Write the operation of multiplying by 0 as a function of $x$ :

$y= \answer {0x}$ .

What operation undoes multiplying by 0? Write this operation as a function of $x$ .

$y=\answer {NA}$

Write the operation of taking a reciprocal as a function of $x$ :

$y= \answer {\frac {1}{x}}$ .

What operation undoes taking a reciprocal? Write this operation as a function of $x$ .

$y=\answer {\frac {1}{x}}$

Write the operation of squaring as a function of $x$ :

$y= \answer {x^2}$ .

What operation undoes squaring? Write this operation as a function of $x$ .

$y=\answer {\sqrt {x}}$

Write the operation of cubing as a function of $x$ :

$y= \answer {x^3}$ .

What operation undoes cubing? Write this operation as a function of $x$ .

$y=\answer {\sqrt [3]{x}}$

Write the operation of raising 10 to a power (e.g., when the input is 2, the output is $10^2$ ) as a function of $x$ :

$y= \answer {10^x}$ .

What operation undoes raising 10 to a power? Write this operation as a function of $x$ .

$y=\answer {\log _{10}(x)}$

When is not possible to undo an operation?

How many of the operations above cannot truly be undone? $\answer {2}$

For each inverse function above, solve that equation for $x$ . What is the relationship between the equation of a function and its inverse?

Solve $1=x+3$ for $x$ .

$x=\answer {-2}$

Solve $-10=x-8$ for $x$ .

$x=\answer {-2}$

Solve $4=-2x$ for $x$ .

$x=\answer {-2}$

Solve $-\frac {1}{3}=\frac {x}{6}$ for $x$ .

$x=\answer {-2}$

Solve $0=0x$ for $x$ .

$x=\answer {NA}$

What is the output of a function that multiplies by 0? Can you “undo” that value without knowing the original input?

If no answer exists, type "NA".

Solve $4=x^2$ for $x$ .

$x=\answer {-2}$

There are multiple answers to this problem. Try inputing

Solve $8=x^3$ for $x$ .

$x=\answer {-2}$

Solve $0.01=10^x$ for $x$ .

$x=\answer {-2}$

How do the steps used to solve these equations relate to solving for the inverse functions you came up with in the previous section?

Which of the equations above have ambiguous answers?

adding 3 subtracting 8 multiplying by $-2$ dividing by 6 multiplying by 0 taking a reciprocal squaring cubing raising $10$ to a power

For each of the operations described above, graph the function and its inverse. (You may want to use https://www.desmos.com/calculator)

What is the relationship between the graph of a function and its inverse?

Examining the graphs above, what property of the function is necessary so that the inverse function produces just one answer? What happens when this property is lacking?