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Chapter 1: Problem 45

Show that the composition of two one-to-one functions is a one-to-one function.

Short Answer

Expert verified
Given two one-to-one functions f and g, we need to show that their composition h(x) = g(f(x)) is also one-to-one. Since f and g are both one-to-one, we know that if f(x) = f(y), then x = y, and if g(x) = g(y), then x = y. Assume h(x) = h(y), then g(f(x)) = g(f(y)). Because g is one-to-one, f(x) = f(y). And as f is one-to-one, x = y. Thus, h is also one-to-one. Therefore, the composition of two one-to-one functions is a one-to-one function.

Step by step solution

01

Define one-to-one functions

A function is one-to-one (injective) if for all distinct inputs, the outputs are also distinct. Mathematically, a function f is one-to-one if, If f(x) = f(y), then x = y.
02

Define the composition of two functions

Given two functions, f and g, their composition is a function h(x) that applies f to the input and then applies g to the result. In mathematical notation, h(x) = g(f(x)).
03

Assume f and g are one-to-one

We are given that both f and g are one-to-one functions. This means that for any x and y, if x 鈮� y, then f(x) 鈮� f(y) and g(x) 鈮� g(y).
04

Prove the composition h is one-to-one

We want to show that for any distinct inputs x and y, h(x) 鈮� h(y). Let's assume h(x) = h(y). Then, g(f(x)) = g(f(y)) Since g is one-to-one, we know that if g(u) = g(v), then u = v. In this case u = f(x) and v = f(y). So, now we have f(x) = f(y). As f is also one-to-one, we can then say that x = y. This means that if h(x) = h(y), then x = y. Therefore, h is a one-to-one function. The composition of two one-to-one functions is a one-to-one function.

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Most popular questions from this chapter

Suppose \(f\) is a function and a function \(g\) is defined by the given expression. (a) Write \(g\) as the composition of \(f\) and one or two linear functions. (b) Describe how the graph of \(g\) is obtained from the graph of \(f\). \( g(x)=2 f(3 x)+4\)

Suppose \(f\) and \(g\) are functions, each with domain of four numbers, with \(f\) and \(g\) defined by the tables below: $$\begin{array}{c|c}x & f(x) \\\\\hline 1 & 4 \\\2 & 5 \\\3 & 2 \\\4 & 3\end{array}$$ $$\begin{array}{c|c}x & g(x) \\\\\hline 2 & 3 \\\3 & 2 \\\4 & 4 \\\5 & 1\end{array}$$ Give the table of values for \(g \circ f\).

Give an example of an increasing function whose domain is the interval [0,1] but whose range does not equal the interval \([f(0), f(1)]\).

Show that if \(f\) is the function defined by \(f(x)=m x+b,\) where \(m \neq 0,\) then \(f\) is a one-toone function.

Show that the composition of two increasing functions is increasing.
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