/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 4 PROBLEMS. Let \(S(x)=x^{2},\) ... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 4

PROBLEMS. Let \(S(x)=x^{2},\) let \(P(x)=2^{x},\) and let \(s(x)=\sin x .\) Find each of the following. In each case you answer should be a mumber. (i) \((S \circ P)(y)\) (ii) \(\quad(S \circ s)(y)\) (iii) \(\quad(S \circ P \circ s)(t)+(s \circ P)(t)\) (iv) \(s\left(t^{3}\right)\)

Short Answer

Expert verified
i) \(2^{2y}\), ii) \((\sin y)^2\), iii) \(2^{2 \sin t} + \sin(2^t)\), iv) \(\sin(t^3)\)

Step by step solution

01

Understand the Notations

Identify what each function represents: - \(S(x) = x^2\) - \(P(x) = 2^x\) - \(s(x) = \sin x\)
02

Calculate \((S \circ P)(y)\)

First, substitute \(P(y)\) into \(S(x)\). This means you evaluate \(S(P(y))\): \(P(y) = 2^y\)\(S(2^y) = (2^y)^2 = 2^{2y}\)So, \((S \circ P)(y) = 2^{2y}\)
03

Calculate \((S \circ s)(y)\)

First, substitute \(s(y)\) into \(S(x)\). This means you evaluate \(S(s(y))\): \(s(y) = \sin y\)\(S(\sin y) = (\sin y)^2\)So, \((S \circ s)(y) = (\sin y)^2\)
04

Calculate \((S \circ P \circ s)(t)\)

First, substitute \(s(t)\) into \(P(x)\) and then substitute that into \(S(x)\): \(s(t) = \sin t\)\(P(\sin t) = 2^{\sin t}\)\(S(2^{\sin t}) = (2^{\sin t})^2 = 2^{2 \sin t}\)So, \((S \circ P \circ s)(t) = 2^{2 \sin t}\)
05

Calculate \((s \circ P)(t)\)

First, substitute \(P(t)\) into \(s(x)\): \(P(t) = 2^t\)\(s(2^t) = \sin(2^t)\)So, \((s \circ P)(t) = \sin(2^t)\)
06

Calculate \((S \circ P \circ s)(t) + (s \circ P)(t)\)

From Step 4, \((S \circ P \circ s)(t) = 2^{2 \sin t}\), and from Step 5, \(s \circ P)(t) = \sin(2^t)\). Adding these together gives:\(2^{2 \sin t} + \sin(2^t)\)
07

Calculate \(s(t^3)\)

Substitute \(t^3\) into \(s(x)\):\(s(t^3) = \sin(t^3)\)

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Key Concepts

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

Composite Functions
A composite function is created when one function is applied to the result of another function. For example, if you have functions \( f(x) \) and \( g(x) \), the composite function \( (f \circ g)(x) \) means you first apply \( g(x) \) and then apply \( f \) to the result. In our exercise:
  • To find \((S \circ P)(y)\), we first evaluate \(P(y) = 2^y\), and then apply \(S\) to that result: \(S(2^y) = (2^y)^2\), which simplifies to \(2^{2y}\).
  • To find \((S \circ s)(y)\), we substitute \(s(y) = \sin y\) into \(S\), resulting in \(S(\sin y) = (\sin y)^2\).
  • For a more complex composition like \((S \circ P \circ s)(t)\), we first find \(s(t) = \sin t\), then \(P(\sin t) = 2^{\sin t}\), and finally apply \(S\) to get \(S(2^{\sin t}) = 2^{2 \sin t}\).
Understanding composite functions helps you break down complex problems into simpler steps.
Exponential Functions
Exponential functions have the general form \(a^x\), where \(a\) is a constant and \(x\) is the variable. These functions can grow very quickly. In our exercise:
  • The function \(P(x) = 2^x\) is an exponential function where the base is 2. When you apply \(P(y) = 2^y\), the output is exponential in nature.
  • When we compose this with another function, like in \((S \circ P)(y)\), the result is \(2^{2y}\), which is still an exponential function but grows even faster.
  • In more complex compositions like \((S \circ P \circ s)(t)\), the intermediate results are also exponential functions, and the complexity increases as more functions are composed.
Remember, exponential functions are powerful tools in modeling growth and decay in various fields.
Trigonometric Functions
Trigonometric functions involve angles and are fundamental in geometry. They are periodic and often used to model cyclical phenomena. For our exercise:
  • The function \(s(x) = \sin x\) represents a basic trigonometric function. It calculates the sine of angle \(x\).
  • In composite functions, \(s\) can be used in interesting ways. For example, in \((S \circ s)(y)\), we first compute \(s(y) = \sin y\) and then square it: \((\sin y)^2\).
  • In \((S \circ P \circ s)(t)\), \(s\) turns the input into a sine value before it's used in other functions. Finally, when finding \(s(t^3)\), we simply compute the sine of \(t^3\): \(\sin(t^3)\).
Mastering trigonometric functions and their compositions is crucial for solving many geometry and physics problems.

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Most popular questions from this chapter

(a) Suppose \(g\) is a function with the property that \(g(x) \neq g(y)\) if \(x \neq y\) Prove that there is a function \(f\) such that \(f \circ g=I\) (b) Suppose that \(f\) is a function such that every number \(b\) can be written \(b=f(a)\) for some number \(a\). Prove that there is a function \(g\) such that \(f \circ g=I\)

(a) If \(x_{1}, \ldots . x_{n}\) are distinct numbers, find a polynomial function \(f_{i}\) of degree \(n-1\) which is 1 at \(x_{i}\) and 0 at \(x_{j}\) for \(j \neq i .\) Hint: the product of all \(\left(x-x_{j}\right)\) for \(j \neq i,\) is 0 at \(x_{j}\) if \(j \neq i .\) (This product is usually denoted by $$ \prod_{j=1 \atop j \neq i}^{n}\left(x-x_{j}\right) $$ the symbol \(\Pi\) (capital pi) playing the same role for products that \(\Sigma\) plays for sums.) (b) Now find a polynomial function \(f\) of degree \(n-1\) such that \(f\left(x_{i}\right)=a_{i}\) where \(a_{1}, \ldots, a_{n}\) are given numbers. (You should use the functions \(f_{i}\) from part (a). The formula you will obtain is called the "Lagrange interpolation formula.")

For which numbers \(a, b, c,\) and \(d\) will the function $$ f(x)=\frac{a x+b}{c x+d} $$ satisfy \(f(f(x))=x\) for all \(x\) (Ior which this equation makes sense)?

(a) Show that \(f=\max (f .0)+\min (f .0) .\) This particular way of writing \(f\) is fairly useful; the functions \(\max (f, 0)\) and \(\min (f, 0)\) are called the positive and negative parts of \(f\). (b) A function \(f\) is called non negative if \(f(x) \geq 0\) for all \(x\). Prove that any function \(f\) can be written \(f=g-h .\) where \(g\) and \(h\) are non negative. in infinitely many ways. (The "standard way" is \(g=\max (f, 0)\) and \(h=\) \(-\min (f .0) .)\) Hint: Any number can certainly be written as the difference of two non negative numbers in infinitely many ways.

(a) Suppose \(f(x)=x+1 .\) Are there any functions \(g\) such that \(f \circ g=g \circ f ?\) (b) Suppose \(f\) is a constant function. For which functions \(g\) does \(f \circ g=\) \(g \circ f ?\) (c) Suppose that \(f \circ g=g\) - \(f\) for all functions \(g\). Show that \(f\) is the identity function, \(f(x)=x\)
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