Find the values of the inverse of a function from its graph; examples and questions are presented along with their detailed solutions and explanations.
If \( f \) is a function whose inverse is \( f^{-1} \), then the relationship between \( f \) and its inverse \( f^{-1} \) is given by:
\[ f(a) = b \iff a = f^{-1}(b) \]
Use the graph of the function \( f \) shown below to find the following values of the inverse function, if possible:
According to the definition of the inverse function:
\[ a = f^{-1}(5) \quad \Leftrightarrow \quad 5 = f(a) \]This means that \( a \) is the value of \( x \) such that \( f(x) = 5 \).
Using the graph below: start from \( y = 5 \) on the y-axis and draw a horizontal line to intersect the graph of \( f \). Then move vertically down to the x-axis to find \( x = 3 \).
Therefore, \( f(3) = 5 \). Hence, \( a = 3 \), and
\[ f^{-1}(5) = 3 \]
\[ a = f^{-1}(0) \iff f(a) = 0 \]
According to the graph shown, \( f(2) = 0 \), and therefore \( f^{-1}(0) = 2 \).
c)
\[ a = f^{-1}(-3) \iff f(a) = -3 \]
The value of \( x \) for which \( f(x) = -3 \) is \( 1 \), and therefore \( f^{-1}(-3) = 1 \).
d)\[ a = f^{-1}(-4) \iff f(a) = -4 \]
The value of \( x \) for which \( f(x) = -4 \) is 0, and therefore \( f^{-1}(-4) = 0 \).
e)\[ a = f^{-1}(-5) \iff f(a) = -5 \]
According to the graph of \( f \), there is no value of \( x \) for which \( f(x) = -5 \), and therefore \( f^{-1}(-5) \) is undefined.
Use the graph of the function \( g \) shown below to find the following values, if possible:
Use the graph of the function \( h \) shown below to find the following, if possible:
According to the definition of the inverse function:
\[ a = g^{-1}(6) \iff g(a) = 6 \]This means that \( a \) is the value of \( x \) such that \( g(x) = 6 \).
Using the graph below, when \( x = 2 \), \( g(x) = 6 \). Hence, \( a = 2 \) and therefore
\[ g^{-1}(6) = 2 \]
\[ a = g^{-1}(0) \iff g(a) = 0 \]
According to the graph, \( g(-1) = 0 \) and therefore
\[ g^{-1}(0) = -1 \] c)\[ a = g^{-1}(-2) \iff g(a) = -2 \]
The value of \( x \) for which \( g(x) = -2 \) is \( -2 \). Therefore,
\[ g^{-1}(-2) = -2 \] d)\[ a = g^{-1}(4) \iff g(a) = 4 \]
The value of \( x \) for which \( g(x) = 4 \) is \( 1 \). Therefore,
\[ g^{-1}(4) = 1 \] e)\[ a = g^{-1}(8) \iff g(a) = 8 \]
According to the graph of \( g \), there is no value of \( x \) such that \( g(x) = 8 \). Therefore,
\[ g^{-1}(8) \text{ is undefined} \]According to the definition of the inverse function:
\[ a = h^{-1}(1) \iff h(a) = 1 \]This means that \( a \) is the value of \( x \) such that \( h(x) = 1 \).
According to the graph shown, \( h(0) = 1 \), and therefore \( h^{-1}(1) = 0 \).
b)\( a = h^{-1}(0) \iff h(a) = 0 \)
According to the graph shown, \( h\left( \frac{\pi}{2} \right) = 0 \), and therefore \( h^{-1}(0) = \frac{\pi}{2} \).
c)\( a = h^{-1}(-1) \iff h(a) = -1 \)
According to the graph shown, \( h(\pi) = -1 \), and therefore \( h^{-1}(-1) = \pi \).
d)\( a = h^{-1}(2) \iff h(a) = 2 \)
According to the graph shown, there is no value of \( x \) for which \( h(x) = 2 \). Therefore, \( h^{-1}(2) \) is undefined.