Chapter 3

When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal shift from a vertical shift?

Can the average rate of change of a function be constant?

Why does the domain differ for different functions?

What is the difference between a relation and a function?

Why do we restrict the domain of the function \(f(x)=x^{2}\) to find the function's inverse?

When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal stretch from a vertical stretch?

If a function \(f\) is increasing on \((a, b)\) and decreasing on \((b, c)\) then what can be said about the local extremum of \(f\) on \((a, c)\) ?

How do we determine the domain of a function defined by an equation?

What is the difference between the input and the output of a function?

Can a function be its own inverse? Explain.

When solving an absolute value function, the isolated absolute value term is equal to a negative number. What does that tell you about the graph of the absolute value function?

When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal compression from a vertical compression?

If the order is reversed when composing two functions, can the result ever be the same as the answer in the original order of the composition? If yes, give an example. If no, explain why not.

How are the absolute maximum and minimum similar to and different from the local extrema?

Explain why the domain of \(f(x)=\sqrt[3]{x}\) is different from the domain of \(f(x)=\sqrt{x}\).

Are one-to-one functions either always increasing or always decreasing? Why or why not?

How can you use the graph of an absolute value function to determine the \(x\) -values for which the function values are negative?

When examining the formula of a function that is the result of multiple transformations, how can you tell a reflection with respect to the \(x\) -axis from a reflection with respect to the \(y\) -axis?

How does the graph of the absolute value function compare to the graph of the quadratic function, \(y=x^{2},\) in terms of increasing and decreasing intervals?

When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket?

How can you determine if a relation is a one-to-one function?

How do you find the inverse of a function algebraically?

Describe all numbers \(x\) that are at a distance of 4 from the number 8 . Express this set of numbers using absolute value notation.

How can you determine whether a function is odd or even from the formula of the function?

Given \(f(x)=x^{2}+2 x\) and \(g(x)=6-x^{2},\) find \(f+g, \quad f-g, \quad f g,\) and \(\quad \frac{f}{g}\)

For the following exercises, find the average rate of change of each function on the interval specified for real numbers \(b\) or \(h\) in simplest form. \(f(x)=4 x^{2}-7\) on \([1, \quad b]\)

Why does the horizontal line test tell us whether the graph of a function is one-to-one?

Show that the function \(f(x)=a-x\) is its own inverse for all real numbers \(a\).

Describe all numbers \(x\) that are at a distance of \(\frac{1}{2}\) from the number -4 . Express this set of numbers using absolute value notation.

For the following exercises, write a formula for the function obtained when the graph is shifted as described. \(f(x)=\sqrt{x}\) is shifted up 1 unit and to the left 2 units.

For the following exercises, determine the domain for each function in interval notation. Given \(f(x)=-3 x^{2}+x\) and \(g(x)=5,\) find \(f+g, \quad f-g, \quad f g,\) and \(\quad \frac{f}{g}\)

For the following exercises, find the domain of each function using interval notation. \(f(x)=-2 x(x-1)(x-2)\)

For the following exercises, determine whether the relation represents a function. \(\\{(a, b), \quad(c, d), \quad(a, c)\\}\)

For the following exercises, find \(f^{-1}(x)\) for each function. \(f(x)=x+3\)

Describe all numbers \(x\) that are at a distance of \(\frac{1}{2}\) from the number -4 . Express this set of numbers using absolute value notation.

For the following exercises, write a formula for the function obtained when the graph is shifted as described. \(f(x)=|x|\) is shifted down 3 units and to the right 1 unit.

For the following exercises, determine the domain for each function in interval notation. Given \(f(x)=2 x^{2}+4 x\) and \(g(x)=\frac{1}{2 x},\) find \(f+g, \quad f-g, \quad f g,\) and \(\frac{f}{g}\)

For the following exercises, find the average rate of change of each function on the interval specified for real numbers \(b\) or \(h\) in simplest form. \(p(x)=3 x+4\) on \([2, \quad 2+h]\)

For the following exercises, find the domain of each function using interval notation. \(f(x)=5-2 x^{2}\)

For the following exercises, determine whether the relation represents a function. \(\\{(a, b),(b, c),(c, c)\\}\)

For the following exercises, find \(f^{-1}(x)\) for each function. \(f(x)=x+5\)

Find all function values \(f(x)\) such that the distance from \(f(x)\) to the value 8 is less than 0.03 units. Express this set of numbers using absolute value notation.

For the following exercises, write a formula for the function obtained when the graph is shifted as described. \(f(x)=|x|\) is shifted down 3 units and to the right 1 unit.

For the following exercises, determine the domain for each function in interval notation. Given \(f(x)=\frac{1}{x-4}\) and \(g(x)=\frac{1}{6-x},\) find \(f+g, \quad f-g, \quad f g,\) and \(\frac{f}{g}\)

For the following exercises, find the average rate of change of each function on the interval specified for real numbers \(b\) or \(h\) in simplest form. \(k(x)=4 x-2\) on \([3, \quad 3+h]\)

For the following exercises, find the domain of each function using interval notation. \(f(x)=3 \sqrt{x-2}\)

For the following exercises, determine whether the relation represents \(y\) as a function of \(x\). \(5 x+2 y=10\)

For the following exercises, find \(f^{-1}(x)\) for each function. \(f(x)=2-x\)

For the following exercises, find the \(x\) - and \(y\) -intercepts of the graphs of each function. \(f(x)=4|x-3|+4\)

For the following exercises, write a formula for the function obtained when the graph is shifted as described. \(f(x)=\frac{1}{x^{2}}\) is shifted up 2 units and to the left 4 units.