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Chapter 2: Problem 40

Find a formula for \(f \circ g\) given the indicated functions \(f\) and \(g\). $$ f(x)=3 x^{-5}, g(x)=-2 x^{-4} $$

Short Answer

Expert verified
The formula for \(f \circ g\) is \(f(g(x)) = \dfrac{3}{32} x^{20}\).

Step by step solution

01

Compute g(x) and Replace x in f(x)

First, we need to check the given functions \(f(x) = 3x^{-5}\) and \(g(x) = -2x^{-4}\). Then, we will compute \(g(x)\) for a general value of x, and replace x with g(x) in the function \(f(x)\). In this case, we have: \(g(x) = -2x^{-4}\) Now, let's replace the x in f(x) with g(x). So, we get: \(f(g(x)) = 3(-2x^{-4})^{-5}\)
02

Simplify the Expression

Next, we will simplify the expression of the composite function: \(f(g(x)) = 3(-2x^{-4})^{-5}\) Using the rule \((ab)^n = a^n b^n\) for exponents, we have: \(f(g(x)) = 3((-2)^{-5} (x^{-4})^{-5})\) Now, we will simplify the exponents: \(f(g(x)) = 3((-2)^{-5} (x^{20}))\) As -2 power -5 equals \(2^{-5}\), we have: \(f(g(x)) = 3(2^{-5} x^{20})\) Finally, we can write the composite function in its simplest form: \(f(g(x)) = \dfrac{3}{32} x^{20}\) So, the formula for \(f \circ g\) is \(f(g(x)) = \dfrac{3}{32} x^{20}\).

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Key Concepts

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

Exponent Rules
When working with exponent rules, it's important to grasp a few basic concepts that help simplify expressions. For example, an exponent represents the number of times to multiply a number by itself. Thus, "squaring" a number means raising it to the power of 2, while "cubing" involves raising it to the power of 3.

Some key exponent rules useful in simplifying expressions include:
  • Product of Powers Rule: When multiplying like bases, add their exponents, i.e., \( a^m \cdot a^n = a^{m+n} \).
  • Power of a Power Rule: When raising an exponent to another power, multiply the exponents, i.e., \( (a^m)^n = a^{m \cdot n} \).
  • Power of a Product Rule: When distributing an exponent over a product, distribute the exponent to each factor, i.e., \( (ab)^n = a^n \cdot b^n \).
  • Negative Exponents: A negative exponent indicates the reciprocal of the base, i.e., \( a^{-n} = \frac{1}{a^n} \).
The exercise involves applying these rules, especially the power of a product rule and negative exponent rule, to simplify the expression \( 3(-2x^{-4})^{-5} \), leading to further simplification.
Simplifying Expressions
Simplifying expressions is all about reducing them to a form that is easier to understand and work with, without changing their value.

Let's take a deeper look at the simplification process used in our exercise. We started with the expression \( 3(-2x^{-4})^{-5} \). To simplify this:
  • First, apply the exponent rule \((ab)^n = a^n b^n\) to separate the parts inside the parentheses: \( (-2)^{-5} \) and \( (x^{-4})^{-5} \).
  • Next, further simplify \( (x^{-4})^{-5} \) by using the power of a power rule, which gives us \( x^{(-4)\cdot(-5)} \) or \( x^{20} \).
  • For \( (-2)^{-5} \), use the negative exponent rule to convert it into the reciprocal form, \( \frac{1}{2^5} \), which evaluates to \( \frac{1}{32} \).
  • Lastly, multiply \(3\) with the simplified terms to get \( \frac{3}{32} x^{20} \).
This step-by-step breakdown simplifies our expression to its most basic and manageable form, which is crucial in many algebraic calculations.
Function Composition
Function composition is a fundamental mathematical operation where you combine two functions to form a new function. Essentially, you're plugging one function into another. This operation is represented by the composition operator "\( \circ \)", and is read as "\( f \) composed with \( g \)."

In our exercise, we are asked to find the composition of the functions \( f(x) = 3x^{-5} \) and \( g(x) = -2x^{-4} \). Here's how it works:
  • Step 1: Identify the functions you are working with. Here, they are \( f(x) \) and \( g(x) \).
  • Step 2: Substitute the output of \( g(x) \) as the input for \( f(x) \). This means replacing every occurrence of \( x \) in \( f(x) \) with \( g(x) \).
  • Step 3: Simplify the resulting expression to achieve the final form of the composite function.
By understanding function composition, you can solve problems where interacting functions need to be evaluated together. It lets you see the effect of one function's output serving as another's input, opening a path to more complex calculations.

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Most popular questions from this chapter

Write each expression as the sum of a polynomial and a rational function whose numerator has smaller degree than its denominator. $$ \frac{x^{6}-4 x^{2}+5}{x^{2}-3 x+1} $$

Find all choices of \(b, c,\) and \(d\) such that -3 and 2 are the only zeros of the polynomial \(p\) defined by $$ p(x)=x^{3}+b x^{2}+c x+d $$.

A new snack shop on campus finds that the number of students following it on Twitter at the end of each of its first five weeks in business is 23,89,223 , \(419,\) and \(647 .\) A clever employee discovers that the number of students following the new snack shop on Twitter after \(w\) weeks is \(p(w),\) where \(p\) is defined by $$p(w)=7+3 w+5 w^{2}+9 w^{3}-w^{4}$$ Indeed, with \(p\) defined as above, we have \(p(1)=23,\) \(p(2)=89, p(3)=223, p(4)=419,\) and \(p(5)=647\) Explain why the polynomial \(p\) defined above cannot give accurate predictions for the number of followers on Twitter for weeks far into the future.

Write each expression as the sum of a polynomial and a rational function whose numerator has smaller degree than its denominator. $$ \frac{4 x-5}{x+7} $$

Verify that \((x+y)^{3}=x^{3}+3 x^{2} y+3 x y^{2}+y^{3}\).
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