Question

How do you find the inverse of $y = - {\left( {\dfrac{1}{3}} \right)^x}$ ?

Answer 1

In the given solution, we have to find the inverse function of the function $y = - {\left( {\dfrac{1}{3}} \right)^x}$ . Inverse of a function exists only if the given function is onto and one-one. In other words, the function must be a bijective function for its inverse to exist. There is a standard method of finding the inverse function of a given function that we will be using in the given problem.

Complete step by step solution:
So to find the inverse function of the given function, we have to first ensure that the function provided to us is a bijection, that is it is both onto and one-one function.
So, we know that the range of exponential functions with any real base is the set of all real numbers. So, the given function is onto. Also, the function $y = - {\left( {\dfrac{1}{3}} \right)^x}$ is a one-one function as it has a unique image in range for every preimage in the domain. Hence, the function is bijective and hence the inverse of the given function exists.
Now, we have to find the inverse function of $y = - {\left( {\dfrac{1}{3}} \right)^x}$ .
So, we have to find the value of x in terms of y.
So, multiplying both sides by $\left( { - 1} \right)$ , we get,
$\Rightarrow {\left( {\dfrac{1}{3}} \right)^x} = - y$
Taking natural logarithm on both sides of the equation, we get,
$\Rightarrow \log {\left( {\dfrac{1}{3}} \right)^x} = \log \left( { - y} \right)$
Now, we know the property of logarithms $\log {x^n} = n\log x$ . So, using this property, we get,
$\Rightarrow x\log \left( {\dfrac{1}{3}} \right) = \log \left( { - y} \right)$
Shifting all the terms except x to right side of the equation in order to find the value of x,
$\Rightarrow x = \dfrac{{\log \left( { - y} \right)}}{{\log \left( {\dfrac{1}{3}} \right)}}$
Now, we have the property of logarithms $\dfrac{{\log x}}{{\log y}} = {\log _y}x$ . Using this property in order to simplify the expression, we get,
$\Rightarrow x = {\log _{\dfrac{1}{3}}}\left( { - y} \right)$
So, we get the inverse function as $f\left( y \right) = {\log _{\dfrac{1}{3}}}\left( { - y} \right)$ .
The inverse function can also be written in terms of x as $f\left( x \right) = {\log _{\dfrac{1}{3}}}\left( { - x} \right)$ by replacing the variable. Hence, the inverse function of the given function $y = - {\left( {\dfrac{1}{3}} \right)^x}$ is $y = {\log _{\dfrac{1}{3}}}\left( { - x} \right)$ .

Note: Also, we know that the inverse function of an exponential function is logarithmic function. So, we can clearly see that the function given to us in the question is exponential function, $y = - {\left( {\dfrac{1}{3}} \right)^x}$ and hence the inverse is a logarithmic function, $y = {\log _{\dfrac{1}{3}}}\left( { - x} \right)$ . We can also verify the answer by going through the solution in the reverse order.