Chapter 1.3, Problem 68E

Solution

a.

To determine which of the twelve basic functions are inverses of each other.

The exponential function and the logarithmic function are the basic functions which are inverses of each other.

Concept Used:

The twelve basic functions:

1)

  

2)

  

3)

  

4)

  

5)

  

6)

  

7)

  

8)

  

9)

  

10)

  

11)

  

12)

  

Calculation:

Observe the natural logarithm function and the exponential function:

  

  

The graph of the exponential function seems to be the reflection of the logarithmic function about $y=x$ and vice versa.

  

Thus, the exponential function and the logarithmic function are the basic functions which are inverses of each other. Also, no any other pair of functions seems to be showing this kind of relationship.

Conclusion:

The exponential function and the logarithmic function are the basic functions which are inverses of each other.

b.

To determine which among the basic functions are inverses of their own.

The identity function and the reciprocal function are the basic functions which are the inverses of their own.

Calculation:

Consider the identity function along with its graph:

  

The graph is that of the line $y=x$ .

Then, reflecting the graph of the identity function about the line $y=x$ yields the graph of the identity function itself.

Thus, the identity function is the inverse of itself.

Now, consider the reciprocal function:

  

The graph of the function remains unchanged even after reflecting it about the line $y=x$ .

Thus, the reciprocal function is also the inverse of itself.

No any other functions seem to show this property.

Thus, the identity function and the reciprocal function are the basic functions which are inverses of their own.

Conclusion:

The identity function and the reciprocal function are the basic functions which are the inverses of their own.

c.

To determine the basic function which is the inverse of another basic function when its domain is restricted to $[0,\infty )$ .

Calculation:

Consider the squaring function:

  

Restrict the domain of the squaring function to $[0,\infty )$:

  

Now, reflect the graph so obtained about the line $y=x$:

  

Thus, the restricted squaring function gives the square root function when it is reflected about the line $y=x$ .

Thus, the basic function which gives the inverse of another basic function when its domain is restricted to $[0,\infty )$ is the squaring function and the corresponding inverse is the square root function.

Conclusion:

The basic function which gives the inverse of another basic function when its domain is restricted to $[0,\infty )$ is the squaring function and the corresponding inverse is the square root function.