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Chapter 8: Problem 33

The following tables represent a function \(f\) that converts cups to quarts and a function \(g\) that converts quarts to gallons (all measurements are for fluids). a. Fill in the missing values in the chart. (Hint: One quart contains 4 cups, and one gallon contains 4 quarts.) b. Now evaluate each of the following and identify the units of the results. i. \((g \circ f)(8)\) iii. \(\left(f^{-1} \circ g^{-1}\right)(1)\) ii. \(g^{-1}(2)\) iv. \(\left(f^{-1} \circ g^{-1}\right)(2)\) c. Explain the significance of \(\left(f^{-1} \circ g^{-1}\right)(x)\) in terms of cups, quarts, and gallons.

Short Answer

Expert verified
i. (g 鈭 f)(8) = 0.5 gallons ii. g鈦宦(2) = 8 quarts iii. (f鈦宦 鈭 g鈦宦)(1) = 16 cups iv. (f鈦宦 鈭 g鈦宦)(2) = 32 cups. (f鈦宦 鈭 g鈦宦)(x) converts gallons to cups by first converting to quarts and then cups.

Step by step solution

01

Understanding the Conversion Factors

Note that 1 quart contains 4 cups, and 1 gallon contains 4 quarts. These will be used to fill in the missing values and convert between units.
02

Fill in the Missing Values in the Tables

Given the relations: 1 quart = 4 cups and 1 gallon = 4 quarts, fill in the missing values for the functions f and g.
03

Table for Function f (Cups to Quarts)

Let f(x) convert cups to quarts. Using the relation 1 quart = 4 cups, complete the table: d & f(d)d = 4:4 cups = 1 quartSo, f(4) = 1 d = 8:8 cups = 2 quartsSo, f(8) = 2 d = 12:12 cups = 3 quartsSo, f(12) = 3 d = 16 (missing):16 cups = 16/4 = 4 quartsSo, f(16) = 4
04

Table for Function g (Quarts to Gallons)

Let g(y) convert quarts to gallons. Using the relation 1 gallon = 4 quarts, complete the table: q & g(q)q = 1:1 quart = 1/4 gallonSo, g(1) = 0.25 q = 2:2 quarts = 2/4 gallonSo, g(2) = 0.5 q = 3:3 quarts = 3/4 gallonSo, g(3) = 0.75 q = 4:4 quarts = 1 gallonSo, g(4) = 1
05

Evaluating (g 鈭 f)(8)

Given: (g 鈭 f)(8) = g(f(8)). From the function f, f(8) = 2 quarts. Now using function g, g(2) = 0.5 gallons.So, (g 鈭 f)(8) = 0.5 gallons.
06

Evaluating g鈦宦(2)

Given: g鈦宦(2). We need to find the value of quarts that convert to 2 gallons. From the inverse function of g, we can deduce g(g鈦宦(x)) = x. So, if g(8) = 2 gallons, then g鈦宦(2) = 8 quarts.
07

Evaluating (f鈦宦 鈭 g鈦宦)(1)

Given: (f鈦宦 鈭 g鈦宦)(1) = f鈦宦(g鈦宦(1)). We need to find g鈦宦(1): From function g, g鈦宦(1) means finding the quarts that become 1 gallon. So, g(4) = 1 gallon thus g鈦宦(1) = 4 quarts. Now, find the cups from 4 quarts: f鈦宦(4). From function f, each quart is 4 cups. So 4 quarts are (4 * 4) = 16 cups. Hence, (f鈦宦 鈭 g鈦宦)(1) = 16 cups.
08

Evaluating (f鈦宦 鈭 g鈦宦)(2)

Given: (f鈦宦 鈭 g鈦宦)(2) = f鈦宦(g鈦宦(2)). We need to find g鈦宦(2): From the inverse function of g, to convert 2 gallons back to quarts: g鈦(8) = 2 (since 4 quarts = 1 gallon, thus 8 quarts = 2 gallons). So, g鈦宦(2) = 8 quarts. Now, f鈦宦(8) means converting 8 quarts to cups. From function f鈦宦, 8 quarts = 8x4 cups = 32 cups. Hence, (f鈦宦 鈭 g鈦宦)(2) = 32 cups.
09

Explain (f鈦宦 鈭 g鈦宦)(x) in Terms of Cups, Quarts, and Gallons

The composition (f鈦宦 鈭 g鈦宦)(x) converts x gallons to the respective number of cups. First, g鈦宦(x) converts x gallons to quarts, then f鈦宦 takes that quart measurement and converts it to the cup measurement.

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Key Concepts

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

Composition of Functions
In mathematics, function composition is the process of applying one function to the results of another. If you have two functions, say \( f \) and \( g \), the composition is written as \( (g \, \circ \, f)(x) \). The notation \( g \, \circ \, f \) means that you first apply \( f \) to \( x \) and then apply \( g \) to the result of \( f(x) \). For example, given the two functions from the exercise:
  • \( f \) converts cups to quarts
  • \( g \) converts quarts to gallons
To find \((g \, \circ \, f)(8)\), you first convert 8 cups to quarts using \( f \), and then convert the resulting quarts to gallons using \( g \). From the tables filled in the solution:
  • \( f(8) = 2 \) quarts
  • \( g(2) = 0.5 \) gallons
Thus, \((g \, \circ \, f)(8) = 0.5\) gallons.
Inverse Functions
An inverse function reverses the effect of the original function. If \( f \) is a function, its inverse \( f^{-1} \) satisfies \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). This means applying a function and then its inverse returns the original value.
  • For \( f(x) \) converting cups to quarts, \( f^{-1}(x) \) will convert quarts back to cups.
  • For \( g(x) \) converting quarts to gallons, \( g^{-1}(x) \) will convert gallons back to quarts.
To evaluate \( (f^{-1} \, \circ \, g^{-1})(1) \), first apply \( g^{-1} \) to 1 gallon to get quarts:
  • \( g^{-1}(1) = 4 \) quarts
Then apply \( f^{-1} \) to those 4 quarts to get cups:
  • \( f^{-1}(4) = 16 \) cups
Hence, \( (f^{-1} \, \circ \, g^{-1})(1) = 16 \) cups.
Fluid Measurement Conversions
Fluid measurements are units used to measure the volume of liquids. The most common conversions involve cups, quarts, and gallons. Understanding these conversions is essential for solving the exercise:
  • 1 quart = 4 cups
  • 1 gallon = 4 quarts
Using these relationships, you can convert between any of these units. For example, to convert 16 cups to quarts:
  • 16 cups \( = \frac{16}{4} = 4 \) quarts
Similarly, to convert quarts to gallons:
  • 4 quarts \( = \frac{4}{4} = 1 \) gallon
These conversions help you fill out the tables and perform the necessary calculations in the exercise.
Evaluation of Functions
Evaluating a function involves plugging in a value for the variable and performing the operations defined by the function. Given function \( f \) that converts cups to quarts and function \( g \) that converts quarts to gallons, here's how you evaluate them:
  • For \( (g \, \circ \, f)(8) \), first calculate \( f(8) \) which gives 2 quarts, then \( g(2) \) which gives 0.5 gallons.
  • For \( g^{-1}(2) \), identify the value of quarts that result in 2 gallons. Since 8 quarts are 2 gallons, \( g^{-1}(2) = 8 \) quarts.
  • For \( (f^{-1} \, \circ \, g^{-1})(1) \), calculate \( g^{-1}(1) \) to get 4 quarts and then \( f^{-1}(4) \) to get 16 cups.
Evaluation shows how inputs are transformed by functions and their inverses, confirming their practical application in unit conversions.

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Most popular questions from this chapter

Let \(f(x)=x^{3}\). a. Write the equation for the new function \(g(x)\) that results from each of the following transformations of \(f(x)\). Explain in words the effect of the transformations. i. \(f(-x)\) iii. \(f(x+2)\) ii. \(-2 f(x)-1\) iv. \(-f(-x)\) b. Sketch by hand the graph of \(f(x)\) and each function in part (a).

(Graphing program optional.) Create a quadratic function in the vertex form \(y=a(x-h)^{2}+k,\) given the specified values for \(a\) and the vertex \((h, k) .\) Then rewrite the function in the standard form \(y=a x^{2}+b x+c .\) If available, use technology to check that the graphs of the two forms are the same. a. \(a=1,(h, k)=(2,-4)\) c. \(a=-2,(h, k)=(-3,1)\) b. \(a=-1,(h, k)=(4,3)\) d. \(a=\frac{1}{2},(h, k)=(-4,6)\)

Find the coordinates of the vertex for each quadratic function listed. Then specify whether each vertex is a maximum or minimum. a. \(y=4 x^{2}\) c. \(P(n)=\left(\frac{1}{12}\right) n^{2}\) b. \(f(x)=-8 x^{2}\) d. \(Q(t)=-\left(\frac{1}{24}\right) t^{2}\)

Let \((h, k)\) be the coordinates of the vertex of a parabola. Then \(h\) equals the average of the two real zeros of the function (if they exist). For each of the following use this to find \(h,\) and then put the equations into the vertex form, \(y=a(x-h)^{2}+k\) a. A parabola with equation \(y=x^{2}+2 x-8\) b. A parabola with equation \(y=-x^{2}-3 x+4\)

For the following quadratic functions in vertex form, \(f(x)=a(x-h)^{2}+k,\) determine the values for \(a, h,\) and \(k\) Then compare each to \(f(x)=x^{2},\) and identify which constants represent a stretch/compression factor, or a shift in a particular direction. a. \(p(x)=5(x-4)^{2}-2\) b. \(g(x)=\frac{1}{3}(x+5)^{2}+4\) c. \(h(x)=-0.25\left(x-\frac{1}{2}\right)^{2}+6\) d. \(k(x)=-3(x+4)^{2}-3\)
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