Chapter 9

If a function \(f\) is given by the formula \(y=f(x),\) then \(f(a)\) is the ________ of \(f\) at \(x=a\)

Find the determinant of the matrix, if it exists. $$\left[\begin{array}{ll} 2 & 0 \\ 0 & 3 \end{array}\right]$$

Write the form of the partial fraction decomposition of the function (as in Example 4). Do not determine the numerical values of the coefficients. $$\frac{1}{(x-1)(x+2)}$$

State whether the equation or system of equations is linear. $$6 x-\sqrt{3} y+\frac{1}{2} z=0$$

Graph the inequality. $$x<3$$

Calculate the products \(A B\) and \(B A\) to verify that \(B\) is the inverse of \(A\). $$A=\left[\begin{array}{ll} 4 & 1 \\ 7 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & -1 \\ -7 & 4 \end{array}\right]$$

Determine whether the matrices \(A\) and \(B\) are equal. $$A=\left[\begin{array}{rrr} 1 & -2 & 0 \\ \frac{1}{2} & 6 & 0 \end{array}\right], \quad B=\left[\begin{array}{rr} 1 & -2 \\ \frac{1}{2} & 6 \end{array}\right]$$

State the dimension of the matrix. $$\left[\begin{array}{rr} 2 & 7 \\ 0 & -1 \\ 5 & -3 \end{array}\right]$$

Use the substitution method to find all solutions of the system of equations. $$\left\\{\begin{aligned}x-y &=2 \\\2 x+3 y &=9\end{aligned}\right.$$

Graph each linear system, either by hand or using a graphing device. Use the graph to determine if the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it. $$\left\\{\begin{aligned} x+y &=4 \\ 2 x-y &=2 \end{aligned}\right.$$

For a function \(f,\) the set of all possible inputs is called the ____________ of \(f,\) and the set of all possible outputs is called the __________ of \(f\).

Find the determinant of the matrix, if it exists. $$\left[\begin{array}{rr} 0 & -1 \\ 2 & 0 \end{array}\right]$$

Write the form of the partial fraction decomposition of the function (as in Example 4). Do not determine the numerical values of the coefficients. $$\frac{x}{x^{2}+3 x-4}$$

State whether the equation or system of equations is linear. $$x^{2}+y^{2}+z^{2}=4$$

Graph the inequality. $$y \geq-2$$

Calculate the products \(A B\) and \(B A\) to verify that \(B\) is the inverse of \(A\). $$A=\left[\begin{array}{ll} 2 & -3 \\ 4 & -7 \end{array}\right], \quad B=\left[\begin{array}{ll} \frac{7}{2} & -\frac{3}{2} \\ 2 & -1 \end{array}\right]$$

Determine whether the matrices \(A\) and \(B\) are equal. $$A=\left[\begin{array}{cc} \frac{1}{4} & \ln 1 \\ 2 & 3 \end{array}\right], \quad B=\left[\begin{array}{cc} 0.25 & 0 \\ \sqrt{4} & \frac{6}{2} \end{array}\right]$$

State the dimension of the matrix. $$\left[\begin{array}{rrrr} -1 & 5 & 4 & 0 \\ 0 & 2 & 11 & 3 \end{array}\right]$$

Use the substitution method to find all solutions of the system of equations. $$\left\\{\begin{array}{r}2 x+y=7 \\\x+2 y=2\end{array}\right.$$

Graph each linear system, either by hand or using a graphing device. Use the graph to determine if the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it. $$\left\\{\begin{aligned} 2 x+y &=11 \\ x-2 y &=4 \end{aligned}\right.$$

(a) Which of the following functions have 5 in their domain? $$f(x)=x^{2}-3 x \quad g(x)=\frac{x-5}{x} \quad h(x)=\sqrt{x-10}$$ (b) For the functions from part (a) that do have 5 in their domain, find the value of the function at 5

Find the determinant of the matrix, if it exists. $$\left[\begin{array}{rr} 4 & 5 \\ 0 & -1 \end{array}\right]$$

State whether the equation or system of equations is linear. $$\left\\{\begin{aligned} x y-3 y+z &=5 \\ x-y^{2}+5 z &=0 \\ 2 x+y z &=3 \end{aligned}\right.$$

Write the form of the partial fraction decomposition of the function (as in Example 4). Do not determine the numerical values of the coefficients. $$\frac{x^{2}-3 x+5}{(x-2)^{2}(x+4)}$$

Graph the inequality. $$y>x$$

Calculate the products \(A B\) and \(B A\) to verify that \(B\) is the inverse of \(A\). $$A=\left[\begin{array}{rrr} 1 & 3 & -1 \\ 1 & 4 & 0 \\ -1 & -3 & 2 \end{array}\right], \quad B=\left[\begin{array}{rrr} 8 & -3 & 4 \\ -2 & 1 & -1 \\ 1 & 0 & 1 \end{array}\right]$$

Perform the matrix operation, or if it is impossible, explain why. $$\left[\begin{array}{rr} 2 & 6 \\ -5 & 3 \end{array}\right]+\left[\begin{array}{rr} -1 & -3 \\ 6 & 2 \end{array}\right]$$

State the dimension of the matrix. $$\left[\begin{array}{l} 12 \\ 35 \end{array}\right]$$

Use the substitution method to find all solutions of the system of equations. $$\left\\{\begin{array}{l}y=x^{2} \\\y=x+12\end{array}\right.$$

Graph each linear system, either by hand or using a graphing device. Use the graph to determine if the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it. $$\left\\{\begin{array}{l} 2 x-3 y=12 \\ -x+\frac{3}{2} y=4 \end{array}\right.$$

A function is given algebraically by the formula \(f(x)=\) \((x-4)^{2}+3 .\) Complete these other ways to represent \(f:\) (a) Verbal: "Subtract \(4,\) then _________ and _________ (b) Numerical: $$\begin{array}{|c|c|} \hline x & f(x) \\ \hline 0 & 19 \\ 2 & \\ 4 & \\ 6 & \\ \hline \end{array}$$

Find the determinant of the matrix, if it exists. $$\left[\begin{array}{rr} -2 & 1 \\ 3 & -2 \end{array}\right]$$

State whether the equation or system of equations is linear. $$\left\\{\begin{aligned} x-2 y+3 z &=10 \\ 2 x+5 y \quad\space\quad &=2 \\ y+2 z &=4 \end{aligned}\right.$$

Write the form of the partial fraction decomposition of the function (as in Example 4). Do not determine the numerical values of the coefficients. $$\frac{1}{x^{4}-x^{3}}$$

Graph the inequality. $$y

Calculate the products \(A B\) and \(B A\) to verify that \(B\) is the inverse of \(A\). $$A=\left[\begin{array}{rrr} 3 & 2 & 4 \\ 1 & 1 & -6 \\ 2 & 1 & 12 \end{array}\right], \quad B=\left[\begin{array}{rrr} 9 & -10 & -8 \\ -12 & 14 & 11 \\ -\frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{array}\right]$$

Perform the matrix operation, or if it is impossible, explain why. $$\left[\begin{array}{lll} 0 & 1 & 1 \\ 1 & 1 & 0 \end{array}\right]-\left[\begin{array}{lll} 2 & 1 & -1 \\ 1 & 3 & -2 \end{array}\right]$$

State the dimension of the matrix. $$\left[\begin{array}{r} -3 \\ 0 \\ 1 \end{array}\right]$$

Use the substitution method to find all solutions of the system of equations. $$\left\\{\begin{aligned} x^{2}+y^{2} &=25 \\ y &=2 x \end{aligned}\right.$$

Graph each linear system, either by hand or using a graphing device. Use the graph to determine if the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it. $$\left\\{\begin{aligned} 2 x+6 y &=0 \\ -3 x-9 y &=18 \end{aligned}\right.$$

Express the rule in function notation. (For example, the rule "square, then subtract 5 " is expressed as the function \(f(x)=x^{2}-5\). Add 3, then multiply by 2

Use back-substitution to solve the triangular system. $$\left\\{\begin{aligned} x-2 y+4 z &=3 \\ y+2 z &=7 \\ z &=2 \end{aligned}\right.$$

Write the form of the partial fraction decomposition of the function (as in Example 4). Do not determine the numerical values of the coefficients. $$\frac{x^{2}}{(x-3)\left(x^{2}+4\right)}$$

Graph the inequality. $$y \leq 2 x+2$$

Perform the matrix operation, or if it is impossible, explain why. $$3\left[\begin{array}{rr} 1 & 2 \\ 4 & -1 \\ 1 & 0 \end{array}\right]$$

State the dimension of the matrix. $$\left[\begin{array}{lll} 1 & 4 & 7 \end{array}\right]$$

Use the substitution method to find all solutions of the system of equations. $$\left\\{\begin{array}{l}x^{2}+y^{2}=8 \\\x+y=0\end{array}\right.$$

Graph each linear system, either by hand or using a graphing device. Use the graph to determine if the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it. $$\left\\{\begin{aligned} -x+\frac{1}{2} y &=-5 \\ 2 x-y &=10 \end{aligned}\right.$$

Express the rule in function notation. (For example, the rule "square, then subtract 5 " is expressed as the function \(f(x)=x^{2}-5\). Divide by \(7,\) then subtract 4

Find the determinant of the matrix, if it exists. $$\left[\begin{array}{l} 3 \\ 0 \end{array}\right]$$