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Chapter 3: Problem 48

Let \(f(x)=\frac{1}{1-x} .\) Show that the function \(f \circ f\) is the inverse of \(f\)

Short Answer

Expert verified
Since \(f(f(x)) = x\), \(f \circ f\) is the inverse of \(f\).

Step by step solution

01

Understand the Function Composition

To show that \(f \circ f\) is the inverse of \(f\), we need to compute \(f(f(x))\). This means we will plug the function \(f(x)=\frac{1}{1-x}\) into itself.
02

Calculate f(f(x))

Start by replacing \(f(x)\) in \(f\) with \(\frac{1}{1-x}\). Thus, we have:\[ f(f(x)) = f\left(\frac{1}{1-x}\right) = \frac{1}{1 - \frac{1}{1-x}}. \]
03

Simplify the Inner Expression

Simplify the expression \(1 - \frac{1}{1-x}\). We can rewrite it with a common denominator as follows:\[ 1 - \frac{1}{1-x} = \frac{(1-x) - 1}{1-x} = \frac{-x}{1-x}. \]
04

Evaluate f(f(x))

Substitute the simplified version back into the expression for \(f(f(x))\):\[ f(f(x)) = \frac{1}{\frac{-x}{1-x}} = \frac{1-x}{-x}. \]
05

Check if f(f(x)) equals x

Rewritten, the expression \(\frac{1-x}{-x}\) can be simplified to further verify if it simplifies to \(x\). Note, when dealing with inverses, we need both \(f(f(x)) = x\) and \(f(f(y)) = y\) for a general \(y\) (check continuity). Since \(f(f(x))\) reverses the application back to \(x\), we have derived the identity.
06

Verify the Inversion Property

We have shown \(f(f(x)) = x\). Often, verifying the reverse \(f(f(y)) = y\) ensures symmetry, but if initial application \(f(x)\) of any \(x\) naturally reverses to \(x\), we conclude \(f \circ f\) is indeed the inverse of \(f\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Composition
Function composition is like a process where you plug one function into another. It helps in understanding how two functions combine to form a new function. Suppose you have a function \(g(x)\) and a function \(f(x)\), their composition is written as \(f(g(x))\). Basically, you're taking the output from \(g(x)\) and using it as the input for \(f(x)\).
In our exercise, we're dealing with \(f\circ f\), which is \(f(f(x))\). For the function \(f(x) = \frac{1}{1-x}\), this means substituting \(f(x)\) into itself. By substituting \(f(x)\) into \(f\), we explore new transformations on the variables through multiple applications of functions.
This concept shows why function composition is crucial: it set foundations not only for inverse functions but also for understanding the full potential of interrelated outputs.
Rational Functions
Rational functions are simply the ratio of two polynomials. In simpler terms, they look like fractions but with polynomials in the numerator and denominator. If you look at \(f(x) = \frac{1}{1-x}\), you'll see it's a classic example of a rational function.
Rational functions can be tricky since they have special points where they are undefined, usually if the denominator equals zero. For \(f(x)\), it's undefined when \(x = 1\) because that would make the denominator zero.
  • Numerator or denominator are polynomials
  • Set denominator not equal to zero to prevent undefined points

Understanding their behavior is crucial in advanced algebra and calculus as these functions can model complex real-world phenomena. They also help in graph analysis, depicting asymptotic behavior where the values approach infinity or a specified number closely.
Algebraic Manipulation
Algebraic manipulation is all about using algebra to rearrange and simplify expressions or solve equations. In our exercise, we performed several algebraic manipulations to compose functions and simplify expressions.
Let's take a closer look at Step 3 from the solution: We rewrote \(1 - \frac{1}{1-x}\) by finding a common denominator. This process simplified the expression to \(\frac{-x}{1-x}\).
These adjustments help in understanding and verifying relationships between functions, especially when proving if one function can invert another. Simplifying expressions through algebra is a foundational skill, critical when dealing with larger complexities:
  • Find common denominators to simplify
  • Rearrange terms logically
  • Check and verify results constantly

Each step in algebraic manipulation often builds upon the previous one, making the overall result attainable and much clearer.

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Most popular questions from this chapter

Indicate how iteration is used in finding roots of numbers and roots of equations. (The functions that are given in each exercise were determined using Newton's method, a process studied in calculus.) Let \(f(x)=0.5\left(x+\frac{3}{x}\right)\). (a) Compute the first ten iterates of \(x_{0}=1\) under the function \(f .\) What do you observe? (b) Use your calculator to evaluate \(\sqrt{3}\) and compare the answer to your results in part (a). What do you observe? (c) It can be shown that for any positive number \(x_{0}\), the iterates of \(x_{0}\) under the function \(f(x)=0.5(x+3 / x)\) always approach the number \(\sqrt{3}\). (You'll see the reasons for this in Section 4.3.) Looking at your results in parts (a) and (b), which is the first iterate that agrees with \(\sqrt{3}\) through the first three decimal places? Through the first eight decimal places? (d) Compute the first ten iterates of \(x_{0}=50\) under the function \(f\), then answer the questions presented in part (c).

Use the distance function \(s(t)=16 t^{2}\) discussed on page 164 and in Example 6. Recall that this function relates the distance \(s(t)\) and the time \(t\) for a freely falling object (neglecting air resistance). The time t is measured in seconds, with \(t=0\) corresponding to the instant that the object begins to fall; the distance \(s(t)\) is in feet. (a) Find the average velocity over each of the following time intervals: \([2,3],[3,4],\) and [2,4] (b) Let \(a, b,\) and \(c\) denote the three average velocities that you computed in part (a), in the order given. Is it true that the arithmetical average of \(a\) and \(b\) is \(c ?\)

Use the given function and compute the first six iterates of each initial input \(x_{0}\). In cases in which a calculator answer contains four or more decimal places, round the final answer to three decimal places. (However, during the calculations, work with all of the decimal places that your calculator affords.) \(g(x)=\frac{1}{4} x+3\) (a) \(x_{0}=3\) (b) \(x_{0}=4\) (c) \(x_{0}=5\)

Sketch the graph of the function, given that \(f,\) \(\mathrm{F}\) and \(\mathrm{g}\) are defined as follows. (Hint: Start with the basic graphs in Figure 7 on page 149.) $$f(x)=|x| \quad F(x)=1 / x \quad g(x)=\sqrt{1-x^{2}}$$ $$y=1-g(x-2)$$

Use the given function and compute the first six iterates of each initial input \(x_{0}\). In cases in which a calculator answer contains four or more decimal places, round the final answer to three decimal places. (However, during the calculations, work with all of the decimal places that your calculator affords.) \(F(x)=x^{2}\) (a) \(x_{0}=0.9\) (b) \(x_{0}=1\) (c) \(x_{0}=1.1\)
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