Chapter 4

Eric said that if \(f(x)=|2-x|\) and \(g(x)=|x-2|,\) then \((f+g)(x)=0 .\) Do you agree with Eric? Explain why or why not.

Is the function \(\mathrm{f}=\\{(x, y) : x y=20\\}\) a polynomial function? Explain why or why not.

Is the set of points on a circle a function? Explain why or why not.

Taylor said that if \((a, b)\) is a pair of a one-to-one function \(f,\) then \((b, a)\) must be a pair of the inverse function \(f^{-1} .\) Do you agree with Taylor? Explain why or why not.

Marcie said that if \(f(x)=x^{2},\) then \(f(a+1)=(a+1)^{2} .\) Do you agree with Marcie? Explain why or why not.

Tiffany said that the polynomial function \(f(x)=x^{4}+x^{2}+1\) cannot have real roots. Do you agree with Tiffany? Explain why or why not.

Megan said that if \(a>1\) and \(g(x)=\frac{1}{a} \mathrm{f}(x),\) then the graph of \(\mathrm{f}(x)\) is the graph of \(\mathrm{g}(x)\) stretched vertically by the factor \(a\) . Do you agree with Megan? Explain why or why not.

If the domain of the function \(f(x)=|3-x|\) is the set of real numbers less than \(3,\) is the function one-to-one? Explain why or why not.

Let \(\mathrm{f}(x)=x^{2}\) and \(\mathrm{g}(x+2)=(x+2)^{2} .\) Are \(\mathrm{f}\) and \(\mathrm{g}\) the same function? Explain why or why not.

Explain why \(\left\\{(x, y) : x=y^{2}\right\\}\) is not a function but \(\\{(x, y) : \sqrt{x}=y\\}\) is a function.

Give an example of a function g for which \(2 \mathrm{g}(x) \neq \mathrm{g}(2 x) .\) Give an example of a function \(\mathrm{f}\) for which \(2 \mathrm{f}(x)=\mathrm{f}(2 x)\)

Explain the difference between direct variation and inverse variation.

Explain why \((x-h)^{2}+(y-k)^{2}=-4\) is not the equation of a circle.

Christopher said that \(\mathrm{f}(x)=|x-2|\) and \(\mathrm{g}(x)=|x+2|\) are inverse functions after he showed that \(\mathrm{f}(\mathrm{g}(2))=2, \mathrm{f}(\mathrm{g}(5))=5,\) and \(\mathrm{f}(\mathrm{g}(7))=7 .\) Do you agree that \(\mathrm{f}\) and \(\mathrm{g}\) are inverse functions? Explain why or why not.

Explain the difference between fg \((x)\) and \(f(g(x))\)

The graph of \(y=x^{2}-4 x+4\) is tangent to the \(x\) -axis at \(x=2\) and does not intersect the \(x\) -axis at any other point. How many roots does this function have? Explain your answer.

Kyle said that if \(r\) is directly proportional to \(s,\) then there is some non- zero constant, \(c,\) such that \(r=c s\) and that \(\\{(s, r)\\}\) is a one-to-one function. Do you agree with Kyle? Explain why or why not.

Eric said that if \(\mathrm{f}(x)=|2-x|,\) then \(y=2-x\) when \(x \leq 2\) and \(y=x-2\) when \(x > 2 . \mathrm{Do}\) you agree with Eric? Explain why or why not.

Let \(f(x)=x^{2}\) and \(g(x+2)=x^{2}+2 .\) Are \(f\) and \(g\) the same function? Explain why or why not.

Can \(y=\sqrt{x}\) define a function from the set of positive integers to the set of positive integers? Explain why or why not.

Let \(f=\\{(0,5),(1,4),(2,3),(3,2),(4,1),(5,0)\\}\) and \(g=\\{(1,1),(2,4),(3,9),(4,16),(5,25)(6,36)\\}\) \(\begin{array}{ll}{\text { a. What is the domain of } \mathrm{f} \text { ? }} & {\text { b. What is the domain of } \mathrm{g} ?} \\ {\text { c. What is the domain of }(\mathrm{g}-\mathrm{f}) ?} & {\text { d. List the ordered pairs of }(\mathrm{g}-\mathrm{f}) \text { in set notation. }} \\ {\text { e. What is the domain of } \frac{\mathrm{g}}{\mathrm{f}} ?} & {\text { f. List the ordered pairs of } \frac{\mathrm{g}}{\mathrm{f}} \text { in set notation. }}\end{array}\)

In \(3-10\) , the coordinates of point \(P\) on the circle with center at \(C\) are given. Write an equation of each circle: a. in center-radius form b. in standard form. $$ P(0,2), C(0,0) $$

In \(3-10,\) find each of the function values when \(\mathrm{f}(x)=4 x\) $$ \mathrm{I}(3) $$

In \(3-10\) , evaluate each composition for the given values if \(f(x)=3 x\) and \(g(x)=x-2\) $$ \mathrm{f}(\mathrm{g}(4)) $$

In \(3-6,\) each set represents a function. a. What is the domain of each function? b. What is the range of each function? c.Is the function one-to-one? $$ \\{(1,4),(2,7),(3,10),(4,13)\\} $$

In \(3-6,\) find the coordinates of the ordered pair with the smallest value of \(y\) for each function. $$ y=|x| $$

In \(3-8,\) for each function: a. Write an expression for \(f(x) .\) b. Find \(f(5)\) \(y=x-2\)

In \(3-5 :\) a. Explain why each set of ordered pairs is or is not a function. b. List the elements of the domain. c. List the clements of the range. $$ \\{(1,1),(2,4),(3,9),(4,16)\\} $$

In \(3-10\) , the coordinates of point \(P\) on the circle with center at \(C\) are given. Write an equation of each circle: a. in center-radius form b. in standard form. $$ P(0,-3), C(0,0) $$

In \(3-10\) , evaluate each composition for the given values if \(f(x)=3 x\) and \(g(x)=x-2\) $$ \mathrm{g}(\mathrm{f}(4)) $$

In \(3-6,\) each set represents a function. a. What is the domain of each function? b. What is the range of each function? c.Is the function one-to-one? $$ \\{(0,8),(2,6),(4,4),(6,2)\\} $$

In \(3-8,\) for each function: a. Write an expression for \(f(x) .\) b. Find \(f(5)\) \(x \stackrel{\mathrm{f}}{\rightarrow} x^{2}\)

In \(3-5 :\) a. Explain why each set of ordered pairs is or is not a function. b. List the elements of the domain. c. List the clements of the range. $$ \\{(1,-1),(0,0),(1,1)\\} $$

In \(3-10\) , the coordinates of point \(P\) on the circle with center at \(C\) are given. Write an equation of each circle: a. in center-radius form b. in standard form. $$ P(4,0), C(0,0) $$

In \(3-10,\) find each of the function values when \(\mathrm{f}(x)=4 x\) $$ \mathrm{I}(-2) $$

In \(3-10\) , evaluate each composition for the given values if \(f(x)=3 x\) and \(g(x)=x-2\) $$ \mathrm{f} \circ \mathrm{g}(-2) $$

In \(3-6,\) each set represents a function. a. What is the domain of each function? b. What is the range of each function? c.Is the function one-to-one? $$ \\{(2,7),(3,7),(4,7),(5,7),(6,7)\\} $$

In \(3-6,\) find the coordinates of the ordered pair with the smallest value of \(y\) for each function. $$ f(x)=\left|\frac{x}{2}-7\right| $$

In \(3-8,\) for each function: a. Write an expression for \(f(x) .\) b. Find \(f(5)\) \(\mathrm{f} : x \rightarrow|3 x-7|\)

In \(3-5 :\) a. Explain why each set of ordered pairs is or is not a function. b. List the elements of the domain. c. List the clements of the range. $$ \\{(-2,5),(-1,5),(0,5),(1,5),(2,5)\\} $$

In \(6-12,\) tell whether the variables vary directly, inversely, or neither. On his way to work, Randy travels 10 miles at \(r\) miles per hour for \(h\) hours.

In \(3-10\) , the coordinates of point \(P\) on the circle with center at \(C\) are given. Write an equation of each circle: a. in center-radius form b. in standard form. $$ P(3,2), C(4,2) $$

In \(3-10,\) find each of the function values when \(\mathrm{f}(x)=4 x\) $$ \mathrm{I}(\mathrm{f}(2)) $$

In \(3-10\) , evaluate each composition for the given values if \(f(x)=3 x\) and \(g(x)=x-2\) $$ g \circ f(-2) $$

In \(3-6,\) each set represents a function. a. What is the domain of each function? b. What is the range of each function? c.Is the function one-to-one? $$ \\{(0,3),(-1,5),(-2,7),(-3,9),(-4,11)\\} $$

In \(3-6,\) find the coordinates of the ordered pair with the smallest value of \(y\) for each function. $$ g(x)=|5-x| $$

In \(3-8,\) for each function: a. Write an expression for \(f(x) .\) b. Find \(f(5)\) \(\\{(x, 5 x)\\}\)

In \(6-12,\) tell whether the variables vary directly, inversely, or neither. A driver travels for 4 hours between stops covering \(d\) miles at a rate of \(r\) miles per hour.

In \(3-10\) , the coordinates of point \(P\) on the circle with center at \(C\) are given. Write an equation of each circle: a. in center-radius form b. in standard form. $$ P(-1,5), C(-1,1) $$

In \(3-10\) , evaluate each composition for the given values if \(f(x)=3 x\) and \(g(x)=x-2\) $$ \mathrm{f}(\mathrm{f}(5)) $$