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Chapter 3: Problem 16

Let \(F(x)=(x+1)^{5}, f(x)=x^{5},\) and \(g(x)=x+1 .\) Which of the following is true for all \(x ?\) \((f \circ g)(x)=F(x) \quad\) or \(\quad(g \circ f)(x)=F(x)\)

Short Answer

Expert verified
\((f \circ g)(x) = F(x)\) is true for all \(x\).

Step by step solution

01

Understanding the Functions and Composition

We have three functions: \( F(x) = (x+1)^5 \), \( f(x) = x^5 \), and \( g(x) = x+1 \). Function composition is applying one function to the results of another, e.g., \((f \circ g)(x) = f(g(x))\). We need to find whether \((f \circ g)(x) = F(x)\) or \((g \circ f)(x) = F(x)\) is true for all \(x\).
02

Checking \((f \circ g)(x)\)

Calculate \((f \circ g)(x)\):\[(f \circ g)(x) = f(g(x)) = f(x+1) = (x+1)^5.\]Since \((f \circ g)(x) = (x+1)^5\), this is equal to \(F(x)\).
03

Checking \((g \circ f)(x)\)

Now, calculate \((g \circ f)(x)\):\[(g \circ f)(x) = g(f(x)) = g(x^5) = x^5 + 1.\]Since \(x^5 + 1 eq (x+1)^5\), \((g \circ f)(x)\) is not equal to \(F(x)\).
04

Conclusion

From our calculations, \((f \circ g)(x) = F(x)\) is true for all \(x\). While \((g \circ f)(x) = F(x)\) is not true for any \(x\).

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Key Concepts

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

Functions
Functions are one of the foundational concepts in mathematics. Think of them as mathematical machines. You feed an input, and they give back an output according to a specific rule.
For example:
  • Consider a function \( f(x) = x^5 \). Here, \( x \) is your input, and the function raises \( x \) to the fifth power as its rule. The output is \( x^5 \).
  • Similarly, another function \( g(x) = x + 1 \) takes \( x \) and simply adds 1 to it. The output now is \( x + 1 \).
Understanding each function's rule is crucial. It helps in determining how they affect the numbers you put into them. In solving problems, you often deal with various functions, which can be combined or composed for more complex operations.
Composite Functions
Composite functions involve combining two or more functions to create a new one. It's like stringing together multiple mathematical machines.
The notation \( (f \circ g)(x) \) represents a composite of functions \( f \) and \( g \). It means you first apply our function \( g \) to \( x \), and then apply function \( f \) to the result.
Let's break it down:
  • Given \( g(x) = x + 1 \) and \( f(x) = x^5 \), composing them results in \( (f \circ g)(x) = f(g(x)) = f(x+1) \).
  • Substitute \( g(x) \) into \( f \). The result is \( f(x+1) = (x+1)^5 \).This is the output of our composite function.
  • Conversely, \( (g \circ f)(x) = g(f(x)) = g(x^5) \) leads to \( x^5 + 1 \), which is different from \( (x+1)^5 \).
Thus, \( (f \circ g)(x) = F(x) \) when \( F(x) = (x+1)^5 \) is accurate, not the other way round.
Function Notation
Function notation is a way to represent and work with functions succinctly. It uses symbols like \( f(x) \), \( g(x) \), and can even be extended with composite symbols like \( (f \circ g)(x) \).
  • The symbol \( f(x) \) indicates a function named \( f \) with \( x \) as its input.
  • Equations like \( f(x) = x^5 \) tell you what the function does to \( x \).
  • In the case of composites like \( (f \circ g)(x) \), it shows the order of operations. Apply \( g(x) \) first, followed by \( f \).
Notation helps to get clear mathematical communication, simplifying the writing of more complex operations. Understanding this notation is crucial in working efficiently with functions, composites, and many other advanced mathematical concepts.

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Most popular questions from this chapter

Use this definition: A prime number is a positive whole number with no factors other than itself and \(1 .\) For example, \(2,13,\) and 37 are primes, but 24 and 39 are not. \(B y\) convention 1 is not considered prime, so the list of the first few primes is as follows: \(2,3,5,7,11,13,17,19,23,29, \ldots\) Let \(G\) be the rule that assigns to each positive integer the nearest prime. For example, \(G(8)=7,\) since 7 is the prime nearest \(8 .\) Explain why \(G\) is not a function. How could you alter the definition of \(G\) to make it a function? Note: There is more than one way to do this.

(a) A function \(f\) is said to be odd if the equation \(f(-x)=-f(x)\) is satisfied by all values of \(x\) in the domain of \(f .\) Show that if \((x, y)\) is a point on the graph of an odd function \(f,\) then the point \((-x,-y)\) is also on the graph. (This implies that the graph of an odd function must be symmetric about the origin.) (b) Show that each function is odd by computing \(f(-x)\) as well as \(-f(x)\) and then noting that the two expressions obtained are equal. (i) \(f(x)=x^{3}\) (iii) \(f(x)=|x| /\left(x+x^{7}\right)\) (ii) \(f(x)=-2 x^{5}+4 x^{3}-x\)

Let \(g(t)=|t-4| .\) Find \(g(3) .\) Find \(g(x+4)\)

Let \(f(x)=(x-a) /(x+a)\) (a) Find \(f(a), f(2 a),\) and \(f(3 a) .\) Is it true that \(f(3 a)=f(a)+f(2 a) ?\) (b) Show that \(f(5 a)=2 f(2 a)\)

Let \(a(x)=1 / x, b(x)=\sqrt[3]{x}, c(x)=2 x+1,\) and \(d(x)=x^{2}\). Express each of the following functions as a composition of two of the given functions. (a) \(f(x)=\sqrt[3]{2 x+1}\) (d) \(K(x)=2 \sqrt[3]{x}+1\) (b) \(g(x)=1 / x^{2}\) (e) \(l(x)=\frac{2}{x}+1\) (c) \(h(x)=2 x^{2}+1\) (f) \(m(x)=\frac{1}{2 x+1}\)
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