Chapter 3

\(29-38=\) Find the maximum or minimum value of the function. $$ f(x)=x^{2}+x+1 $$

A function \(f\) is given. (a) Use a graphing calculator to draw the graph of \(f .\) (b) Find the domain and range of \(f\) from the graph. $$ f(x)=4 $$

\(27-32\) : A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=\sqrt{x} ;\) shift 3 units to the left, stretch vertically by a factor of \(5,\) and reflect in the \(x\) -axis

Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other. \(f(x)=\frac{1}{x-1}, \quad x \neq 1\) \(g(x)=\frac{1}{x}+1, \quad x \neq 0\)

\(29-30\) A linear function is given. (a) Find the average rate of change of the function between \(x=a\) and \(x=a+h .\) (b) Show that the average rate of change is the same as the slope of the line. $$ f(x)=\frac{1}{2} x+3 $$

Find \(f(a), f(a+h),\) and the difference quotient \(\frac{f(a+h)-f(a)}{h},\) where \(h \neq 0\). $$ f(x)=3 x+2 $$

\(29-40\) Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=6 x-5, \quad g(x)=\frac{x}{2} $$

A function \(f\) is given. (a) Use a graphing calculator to draw the graph of \(f .\) (b) Find the domain and range of \(f\) from the graph. $$ f(x)=-x^{2} $$

\(27-32\) : A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=\sqrt[3]{x}\) , reflect in the \(y\) -axis, shrink vertically by a factor of \(\frac{1}{2},\) and shift upward \(\frac{1}{3}\) unit

\(29-30\) A linear function is given. (a) Find the average rate of change of the function between \(x=a\) and \(x=a+h .\) (b) Show that the average rate of change is the same as the slope of the line. $$ g(x)=-4 x+2 $$

Find \(f(a), f(a+h),\) and the difference quotient \(\frac{f(a+h)-f(a)}{h},\) where \(h \neq 0\). $$ f(x)=x^{2}+1 $$

\(29-40\) Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=x^{2}, \quad g(x)=x+1 $$

\(29-38=\) Find the maximum or minimum value of the function. $$ f(t)=100-49 t-7 t^{2} $$

A function \(f\) is given. (a) Use a graphing calculator to draw the graph of \(f .\) (b) Find the domain and range of \(f\) from the graph. $$ f(x)=4-x^{2} $$

\(27-32\) : A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=|x| ;\) shift to the right \(\frac{1}{2}\) unit, shrink vertically by a factor of \(0.1,\) and shift downward 2 units

Find the inverse function of \(f\). \(f(x)=2 x+1\)

Find \(f(a), f(a+h),\) and the difference quotient \(\frac{f(a+h)-f(a)}{h},\) where \(h \neq 0\). $$ f(x)=5 $$

\(29-40\) Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=x^{3}+2, \quad g(x)=\sqrt[3]{x} $$

\(29-38=\) Find the maximum or minimum value of the function. $$ f(t)=10 t^{2}+40 t+113 $$

A function \(f\) is given. (a) Use a graphing calculator to draw the graph of \(f .\) (b) Find the domain and range of \(f\) from the graph. $$ f(x)=x^{2}+4 $$

\(27-32\) : A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=|x| ;\) shift to the left 1 unit, stretch vertically by a factor of \(3,\) and shift upward 10 units

Find the inverse function of \(f\). \(f(x)=6-x\)

Find \(f(a), f(a+h),\) and the difference quotient \(\frac{f(a+h)-f(a)}{h},\) where \(h \neq 0\). $$ f(x)=\frac{1}{x+1} $$

\(29-40\) Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=\frac{1}{x}, \quad g(x)=2 x+4 $$

\(29-38=\) Find the maximum or minimum value of the function. $$ f(s)=s^{2}-1.2 s+16 $$

A function \(f\) is given. (a) Use a graphing calculator to draw the graph of \(f .\) (b) Find the domain and range of \(f\) from the graph. $$ f(x)=\sqrt{16-x^{2}} $$

33–48 ? Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ f(x)=(x-2)^{2} $$

Find the inverse function of \(f\). \(f(x)=4 x+7\)

Find \(f(a), f(a+h),\) and the difference quotient \(\frac{f(a+h)-f(a)}{h},\) where \(h \neq 0\). $$ f(x)=\frac{x}{x+1} $$

\(29-40\) Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=x^{2}, \quad g(x)=\sqrt{x-3} $$

\(29-38=\) Find the maximum or minimum value of the function. $$ g(x)=100 x^{2}-1500 x $$

A function \(f\) is given. (a) Use a graphing calculator to draw the graph of \(f .\) (b) Find the domain and range of \(f\) from the graph. $$ f(x)=-\sqrt{25-x^{2}} $$

33–48 ? Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ f(x)=(x+7)^{2} $$

Find the inverse function of \(f\). \(f(x)=3-5 x\)

Find \(f(a), f(a+h),\) and the difference quotient \(\frac{f(a+h)-f(a)}{h},\) where \(h \neq 0\). $$ f(x)=\frac{2 x}{x-1} $$

\(29-40\) Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=|x|, \quad g(x)=2 x+3 $$

\(29-38=\) Find the maximum or minimum value of the function. $$ h(x)=\frac{1}{2} x^{2}+2 x-6 $$

A function \(f\) is given. (a) Use a graphing calculator to draw the graph of \(f .\) (b) Find the domain and range of \(f\) from the graph. $$ f(x)=\sqrt{x-1} $$

33–48 ? Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ f(x)=-(x+1)^{2} $$

Find the inverse function of \(f\). \(f(x)=\frac{x}{2}\)

Find \(f(a), f(a+h),\) and the difference quotient \(\frac{f(a+h)-f(a)}{h},\) where \(h \neq 0\). $$ f(x)=3-5 x+4 x^{2} $$

\(29-40\) Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=x-4, \quad g(x)=|x+4| $$

\(29-38=\) Find the maximum or minimum value of the function. $$ f(x)=-\frac{x^{2}}{3}+2 x+7 $$

A function \(f\) is given. (a) Use a graphing calculator to draw the graph of \(f .\) (b) Find the domain and range of \(f\) from the graph. $$ f(x)=\sqrt{x+2} $$

33–48 ? Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ f(x)=1-x^{2} $$

Find the inverse function of \(f\). \(f(x)=\frac{1}{x^{2}}, \quad x>0\)

Find \(f(a), f(a+h),\) and the difference quotient \(\frac{f(a+h)-f(a)}{h},\) where \(h \neq 0\). $$ f(x)=x^{3} $$

\(29-40\) Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=\frac{x}{x+1}, \quad g(x)=2 x-1 $$

\(29-38=\) Find the maximum or minimum value of the function. $$ f(x)=3-x-\frac{1}{2} x^{2} $$

Sketch the graph of the piecewise defined function. $$ f(x)=\left\\{\begin{array}{ll}{0} & {\text { if } x<2} \\ {1} & {\text { if } x \geq 2}\end{array}\right. $$