/*! 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 2 Let \(\mathbb{R}^{*}=\\{x \in \m... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 2

Let \(\mathbb{R}^{*}=\\{x \in \mathbb{R} \mid x \geq 0\\},\) and let \(s: \mathbb{R} \rightarrow \mathbb{R}^{*}\) be defined by \(s(x)=x^{2}\) (a) Evaluate \(s(-3), s(-1), s(1),\) and \(s(3)\). (b) Determine the set of all of the preimages of 0 and the set of all preimages of 2 (c) Sketch a graph of the function \(s\). (d) Determine the range of the function \(s\).

Short Answer

Expert verified
(a) The evaluations of s(x) are: s(-3) = 9, s(-1) = 1, s(1) = 1, s(3) = 9. (b) The preimages of 0 are {0}, and the preimages of 2 are {鈭掆垰2, 鈭�2}. (c) The graph of s(x) = x^2 is a parabola with its vertex at the origin (0,0) and opening upwards. (d) The range of the function s is "\( \mathbb{R}^{*} = \{ x \in \mathbb{R} \mid x \geq 0 \}\)".

Step by step solution

01

(a) Evaluating s(x) at different values of x

To evaluate s(x) at different given values of x, plug the values into the definition of s(x), and compute the result for each of the specified x values: s(-3) = (-3)^2 = 9 s(-1) = (-1)^2 = 1 s(1) = (1)^2 = 1 s(3) = (3)^2 = 9
02

(b) Finding preimages of 0 and 2

To find the preimages of 0 and 2, we want to find the values of x that satisfy s(x) = 0 and s(x) = 2 respectively. For s(x) = 0: x^2 = 0 x = 0 For s(x) = 2: x^2 = 2 x = 卤\sqrt{2} Therefore, the preimages of 0 are {0}, and the preimages of 2 are {鈭掆垰2, 鈭�2}.
03

(c) Sketching the graph of s(x) = x^2

To sketch the graph of s(x) = x^2, we can plot a few points on the graph and observe its general shape. We have already found a few function values: s(-3) = 9 s(-1) = 1 s(0) = 0 s(1) = 1 s(3) = 9 These points lie on a parabola with its vertex at the origin (0, 0) and opening upwards. Connect these points to obtain the sketch of the graph of s(x) = x^2.
04

(d) Determining the range of s(x) = x^2

The range of a function is the set of all possible output values. Since s(x) = x^2 for every real number x, the smallest possible value of s(x) is 0, which occurs at x = 0. However, s(x) can take on any positive value as well. To see this, notice that for any positive number y, we have an x (namely, 鈭歽) such that s(x) = y: s(鈭歽) = (鈭歽)^2 = y Thus, the range of the function s is "\( \mathbb{R}^{*} = \{ x \in \mathbb{R} \mid x \geq 0 \}\)".

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Key Concepts

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

Preimages and Images
When dealing with functions, understanding preimages and images is crucial. Think of a function as a machine that takes inputs (preimages) and transforms them into outputs (images). For instance, in the function defined by \(s(x) = x^2\), if we input a number like \(-3\), the function will map it to an image \(9\) since \((-3)^2 = 9\).
Here, \(-3\) is the preimage and \(9\) is the image.To determine the preimages of some output value, we reverse this process. For \(s(x) = 0\), the preimage is \(0\) because \(0^2 = 0\). Similarly, for \(s(x) = 2\), we solve the equation \(x^2 = 2\). This gives us preimages \(\pm\sqrt{2}\).
  • Preimage: Input value that a function maps to an output.
  • Image: Output value generated by a function from an input.
Knowing the preimages and images helps us comprehend the behavior and mapping of a function.
Graph of a Function
Visualizing functions through graphs can be a powerful tool to understand their behavior. The graph of the function \(s(x) = x^2\) is parabolic, meaning it has a U-shape. This shape is symmetric around the y-axis with a vertex at \((0,0)\).
It provides a quick visual reference for the function's value at any point.Plotting a few points can help sketch the graph accurately. For example, place points at \((-3, 9)\), \((-1, 1)\), \((0, 0)\), \((1, 1)\), and \((3, 9)\). Connecting these dots will give you the upward-opening parabola.
  • Vertex: The lowest point on the graph of \(s(x) = x^2\) is \((0,0)\).
  • Symmetry: Reflective symmetry about the y-axis.
Graphs offer insights into the trends and potential outputs of functions across different inputs.
Range of a Function
The range of a function is all the possible outcomes it can produce. For the function \(s(x) = x^2\), this means determining all potential values \(s(x)\) can reach.
Since squaring any real number results in a non-negative value, the range consists of non-negative real numbers.The smallest possible value for \(s(x)\) is \(0\), occurring at \(x = 0\). As \(x\) increases or decreases, all positive numbers are accessible in this function, as each positive \(y\) has a corresponding \(x\) (denoted \(\pm\sqrt{y}\) such that \(s(x) = y\)).
  • Minimum of Range: The smallest possible output is \(0\).
  • Comprehensive: It spans all non-negative values.
Thus, the range of \(s\) is expressed as \(\mathbb{R}^{*} = \{ x \in \mathbb{R} \mid x \geq 0 \}\), illustrating the function鈥檚 potential outcomes.

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Most popular questions from this chapter

Prove Part (2) of Corollary 6.28. Let \(A\) and \(B\) be nonempty sets and let \(f: A \rightarrow B\) be a bijection. Then for every \(y\) in \(B,\left(f \circ f^{-1}\right)(y)=y\).

(a) Let \(S=\\{1,2,3,4\\} .\) Define \(F: S \rightarrow \mathbb{N}\) by \(F(x)=x^{2}\) for each \(x \in S\). What is the range of the function \(F\) and what is \(F(S) ?\) How do these two sets compare? Now let \(A\) and \(B\) be sets and let \(f: A \rightarrow B\) be an arbitrary function from \(A\) to \(B\). (b) Explain why \(f(A)=\operatorname{range}(f)\). (c) Define a function \(g: A \rightarrow f(A)\) by \(g(x)=f(x)\) for all \(x\) in \(A\). Prove that the function \(g\) is a surjection.

Let \(f:(\mathbb{R}-\\{0\\}) \rightarrow \mathbb{R}\) by \(f(x)=\frac{x^{3}+5 x}{x}\) and let \(g: \mathbb{R} \rightarrow \mathbb{R}\) by \(g(x)=x^{2}+5\) (a) Calculate \(f(2), f(-2), f(3),\) and \(f(\sqrt{2})\) (b) Calculate \(g(0), g(2), g(-2), g(3),\) and \(g(\sqrt{2})\) (c) Is the function \(f\) equal to the function \(g ?\) Explain. (d) Now let \(h:(\mathbb{R}-\\{0\\}) \rightarrow \mathbb{R}\) by \(h(x)=x^{2}+5\). Is the function \(f\) equal to the function \(h\) ? Explain.

Recall that a real function is a function whose domain and codomain are subsets of the real numbers \(\mathbb{R}\). (See page 288.) Most of the functions used in calculus are real functions. Quite often, a real function is given by a formula or a graph with no specific reference to the domain or the codomain. In these cases, the usual convention is to assume that the domain of the real function \(f\) is the set of all real numbers \(x\) for which \(f(x)\) is a real number, and that the codomain is \(\mathbb{R}\). For example, if we define the (real) function \(f\) by $$ f(x)=\frac{x}{x-2} $$ we would be assuming that the domain is the set of all real numbers that are not equal to 2 and that the codomain is \(\mathbb{R}\). Determine the domain and range of each of the following real functions. It might help to use a graphing calculator to plot a graph of the function. (a) The function \(k\) defined by \(k(x)=\sqrt{x-3}\) (b) The function \(F\) defined by \(F(x)=\ln (2 x-1)\) (c) The function \(f\) defined by \(f(x)=3 \sin (2 x)\) (d) The function \(g\) defined by \(g(x)=\frac{4}{x^{2}-4}\) (e) The function \(G\) defined by \(G(x)=4 \cos (\pi x)+8\)

Let \(f: S \rightarrow T\) and let \(C\) and \(D\) be subsets of \(T\). Prove or disprove each of the following: (a) If \(C \subseteq D,\) then \(f^{-1}(C) \subseteq f^{-1}(D)\) (b) If \(f^{-1}(C) \subseteq f^{-1}(D),\) then \(C \subseteq D\)
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