Find The Inverse Function from Tables
Questions With Solutions

Find the values of the inverse of a function given by a table? Questions are presented along with detailed Solutions and explanations.


Relationship Between a Function and its Inverse

If \( f \) is a function whose inverse is \( f^{-1} \), then the relationship between \( f \) and \( f^{-1} \) is written as:

\[ f(a) = b \iff a = f^{-1}(b) \]

Examples

Use the table below to find the following if possible:
a) f -1(- 4) ,   b) f -1(6) ,   c) f -1(9) ,   d) f -1(10) ,   e) f -1(-10)

Use the table below to find the following if possible:

\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -4 & 9 \\ \hline -2 & 4 \\ \hline 0 & -5 \\ \hline 3 & 5 \\ \hline 6 & -4 \\ \hline 7 & 7 \\ \hline 8 & -10 \\ \hline \end{array} \]

Solution

a)

According to the definition of the inverse function:

\[ a = f^{-1}(-4) \iff f(a) = -4 \]

This means that \( a \) is the value of \( x \) such that \( f(x) = -4 \). From the given table, for \( x = 6 \), \( f(x) = -4 \). Hence:

\[ a = 6 \quad \text{and therefore} \quad f^{-1}(-4) = 6 \]

b)

\[ a = f^{-1}(6) \iff f(a) = 6 \]

There is no value of \( x \) for which \( f(x) = 6 \), and therefore \( f^{-1}(6) \) is undefined.

c)

\[ a = f^{-1}(9) \iff f(a) = 9 \]

The value of \( x \) for which \( f(x) = 9 \) is \( x = -4 \). Therefore:

\[ f^{-1}(9) = -4 \]

d)

\[ a = f^{-1}(10) \iff f(a) = 10 \]

There is no value of \( x \) for which \( f(x) = 10 \), and therefore \( f^{-1}(10) \) is undefined.

e)

\[ a = f^{-1}(-10) \iff f(a) = -10 \]

The value of \( x \) for which \( f(x) = -10 \) is \( x = 8 \), and therefore:

\[ f^{-1}(-10) = 8 \]

More Questions with Solutions

Use the table below to find the following if possible:

  1. \( g^{-1}(0) \)
  2. \( g^{-1}(-10) \)
  3. \( g^{-1}(-5) \)
  4. \( g^{-1}(-7) \)
  5. \( g^{-1}(3) \)
\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -5 & 9 \\ \hline -2 & 3 \\ \hline 0 & -5 \\ \hline 3 & 7 \\ \hline 4 & -4 \\ \hline 7 & 10 \\ \hline 11 & 0 \\ \hline \end{array} \]

Solution

a)

According to the definition of the inverse function:
\[ a = g^{-1}(0) \iff g(a) = 0 \] This means that \( a \) is the value of \( x \) such that \( g(x) = 0 \).
Using the table above, for \( x = 11 \), we have \( g(x) = 0 \).
Hence, \( a = 11 \), and therefore \[ g^{-1}(0) = 11 \]

b)

\( a = g^{-1}(-5) \) if and only if \( g(a) = -5 \).
The value of \( x \) for which \( g(x) = -5 \) is \( 0 \).
Therefore, \[ g^{-1}(-5) = 0 \]

c)

\( a = g^{-1}(-10) \) if and only if \( g(a) = -10 \).
There is no value of \( x \) for which \( g(x) = -10 \).
Therefore, \[ g^{-1}(-10) \; \text{ is undefined} \]

d)

\( a = g^{-1}(-7) \iff g(a) = -7 \).
There is no value of \( x \) for which \( g(x) = -7 \).
Therefore, \[ g^{-1}(-7) \; \text{ is undefined} \]

e)

\( a = g^{-1}(3) \) if and only if \( g(a) = 3 \).
The value of \( x \) for which \( g(x) = 3 \) is \( -2 \).
Therefore, \[ g^{-1}(3) = -2 \]

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