a. Let m be any positive integer, and define f (x) = xm for all nonnegative real numbers

The graph of a function f is shown below. a. Is f(0) positive or negative? b. For what values of x does f(x) = 0? c. Find approximate values for \(x_{1}\) and \(x_{2}\) so that \(f\left(x_{1}\right)=f\left(x_{2}\right)=1\) but \(x_{1} \neq x_{2}\). d. Find an approximate value for x such that f(x) = 1.5. e. As x increases from -3 to -1, do the values of f increase or decrease? f. As x increases from 0 to 4 , do the values of f increase or decrease? Text Transcription: x_1 x_2 f(x_1) = f(x_2) = 1 X_1 neq x_2

The graph of a function g is shown below. a. Is g(0) positive or negative? b. Find an approximate value of x so that g(x) = 0. c. Find approximate values for \(x_{1}\) and \(x_{2}\) so that \(g\left(x_{1}\right)=\) \(g\left(x_{2}\right)=1\) but \(x_{1} \neq x_{2}\). d. Find an approximate value for x such that g(x) = -2. e. As x increases from -2 to 1 , do the values of g increase or decrease? f. As x increases from 1 to 3 , do the values of g increase or decrease? Text Transcription: x_1 x_2 g(x_1)= g(x_2)=1 x_1 neq x_2

Draw the graphs of the power functions \(p_{1 / 3}\) and \(p_{1 / 4}\) on the same set of axes. When 0 < x < 1, which is greater: \(x^{1 / 3}\) or \(x^{1 / 4}\)? When x > 1, which is greater: \(x^{1 / 3}\) or \(x^{1 / 4}\)? Text Transcription: p_1/3 p_1/4 x^1/3 x^1/4

Draw the graphs of the power functions \(p_{3}\) and \(p_{4}\) on the same set of axes. When 0 < x < 1, which is greater: \(x^{3}\) or \(x^{4}\)? When x > 1, which is greater: \(x^{3}\) or \(x^{4}\)? Text Transcription: p_3 p_4 x^3 x^4

Draw the graphs of \(y=2\lfloor x\rfloor\) and \(y=\lfloor 2 x\rfloor\) for all real numbers x. What can you conclude from these graphs? Text Transcription: y=2\lfloor x\rfloor

Graph each of the functions defined in 6-9 below. \(g(x)=\lceil x\rceil\) for all real numbers x (Recall that the ceiling of \(x,\lceil x\rceil\), is the least integer that is greater than or equal to x. That is, \(\lceil x\rceil=\) the unique integer n such that \(n-1<x \leq n\).) Text Transcription: g(x)=\lceil x\rceil x,\lceil x\rceil

Graph each of the functions defined in 6-9 below. \(h(x)=\lceil x\rceil-\lfloor x\rfloor\) for all real numbers x Text Transcription: h(x)=\lceil x\rceil-\lfloor x\rfloor

Graph each of the functions defined in 6-9 below. \(F(x)=\left\lfloor x^{1 / 2}\right\rfloor\) for all real numbers x Text Transcription: F(x)=lfloor x^1 / 2rfloor

Graph each of the functions defined in 6-9 below. \(G(x)=x-\lfloor x\rfloor\) for all real numbers x Text Transcription: G(x)=x-\lfloor x\rfloor

In each of 10-13 a function is defined on a set of integers. Graph each function. \(f(n)=|n|\) for each integer n Text Transcription: f(n)=|n|

In each of 10-13 a function is defined on a set of integers. Graph each function. g(n) = (n / 2) + 1 for each integer n

In each of 10-13 a function is defined on a set of integers. Graph each function. \(h(n)=\lfloor n / 2\rfloor\) for each integer \(n \geq 0\) Text Transcription: h(n)=\lfloor n / 2\rfloor n \geq 0

In each of 10-13 a function is defined on a set of integers. Graph each function. \(k(n)=\left\lfloor n^{1 / 2}\right\rfloor\) for each integer \(n \geq 0\) Text Transcription: k(n)=lfloor n^1 / 2rfloor n \geq 0

The graph of a function f is shown below. Find the intervals on which f is increasing and the intervals on which f is decreasing.

Show that the function \(f: \mathbf{R} \rightarrow \mathbf{R}\) defined by the formula f(x) = 2x - 3 is increasing on the set of all real numbers. Text Transcription: f: R rightarrow R

Show that the function \(g: \mathbf{R} \rightarrow \mathbf{R}\) defined by the formula g(x) = -(x / 3) + 1 is decreasing on the set of all real numbers. Text Transcription: g: z R rightarrow R

Let h be the function from R to R defined by the formula \(h(x)=x^{2}\) for all real numbers x. a. Show that h is decreasing on the set of all real numbers less than zero. b. Show that h is increasing on the set of all real numbers greater than zero. Text Transcription: h(x)=x^2

Let \(k: \mathbf{R} \rightarrow \mathbf{R}\) be the function defined by the formula k(x) = (x-1) / x for all real numbers \(x \neq 0\). a. Show that k is increasing for all real numbers x > 0. b. Is k increasing or decreasing for x < 0? Prove your answer. Text Transcription: k:R rightarrow R x \neq 0

Show that if a function \(f: \mathbf{R} \rightarrow \mathbf{R}\) is increasing, then f is one-to-one. Text Transcription: f: R rightarrow R

Given real-valued functions f and g with the same domain D, the sum of f and g, denoted f + g, is defined as follows: For all real numbers x, (f + g)(x) = f(x) + g(x). Show that if f and g are both increasing on a set S, then f + g is also increasing on S.

a. Let m be any positive integer, and define \(f(x)=x^{m}\) for all nonnegative real numbers x. Use the binomial theorem to show that f is an increasing function. b. Let m and n be any positive integers, and let \(g(x)=x^{m / n}\) for all nonnegative real numbers x. Prove that g is an increasing function. The results of this exercise are used in the exercises for Sections 11.2 and 11.4. Text Transcription: f(x)=x^m g(x)=x^m / n

Let f be the function whose graph is shown below. Draw the graph of 3f .

Let h be the function whose graph is shown below. Draw the graph of 2h.

Let f be a real-valued function of a real variable. Show that if f is decreasing on a set S and if M is any positive real number, then M f is decreasing on S.

Let f be a real-valued function of a real varaible. Show that if f is increasing on a set S and if M is any negative real number, then M f is decreasing on S.

Let f be a real-valued function of a real variable. Show that if f is decreasing on a set S and if M is any negative real number, then M f is increasing on S.

In 27 and 28 , functions f and g are defined. In each case draw the graphs of f and 2g on the same set of axes and find a number \(x_{0}\) so that \(f(x) \leq 2 g(x)\) for all \(x>x_{0}\). You can find an exact value for \(x_{0}\) by solving a quadratic equation, or you can find an approximate value for \(x_{0}\) by using a graphing calculator. \(f(x)=x^{2}+10 x+11\) and \(g(x)=x^{2}\) for all real numbers \(x \geq 0\) Text Transcription: x_0 f(x) \leq 2 g(x) x>x_0 f(x)=x^2+10 x+11 g(x)=x^2 x geq 0

In 27 and 28 , functions f and g are defined. In each case draw the graphs of f and 2g on the same set of axes and find a number \(x_{0}\) so that \(f(x) \leq 2 g(x)\) for all \(x>x_{0}\). You can find an exact value for \(x_{0}\) by solving a quadratic equation, or you can find an approximate value for \(x_{0}\) by using a graphing calculator. \(f(x)=x^{2}+125 x+254\) and \(g(x)=x^{2}\) for all real numbers \(x \geq 0\) Text Transcription: x_0 f(x) \leq 2 g(x) x>x_0 f(x)=x^2+125 x+254 g(x)=x^2 x \geq 0