Chapter 1

Find an equation for the line which is tangent to the circle \(x^{2}+y^{2}-2 x+6 y-15=0\) at the point \((4,1) .\) HINT: \(A\) line is tangent to a circle at a point \(P\) iff it is perpendicular to the radius at \(P\)

Determine all numbers \(A-0\) for which the statement is true. $$\text { If }|x+1| < 2, \text { then }|3 x+3| < A$$.

Sketch the graph of the function. $$f(x)=3 \cos 2 x$$

State whether the set is bounded above, bounded below, bounded. If a set is bounded above, give an upper bound; if it is bounded below, give a lower bound; if it is bounded, give an upper bound and a lower bound. The set of rational numbers less than \(\sqrt{2}\).

Find the solutions \(x\) that are in the interval [0.2 \(\pi\) ]. Express your answers in radians and use four decimal place accuracy. $$\tan x=6.7192$$.

The point \(P(1,-1)\) is on a circle centered at \(C(-1,3) .\) Find an equation for the line tangent to the circic at \(P\).

Arrange the following in order: \(1, x, \sqrt{x}, 1 / x .1 / \sqrt{x}\). given that: \((a) x>1 ;(b) 0 < x < 1\).

Sketch the graph of the function. $$f(x)=\frac{1}{3} \cos 2 x$$

Find the solutions \(x\) that are in the interval [0.2 \(\pi\) ]. Express your answers in radians and use four decimal place accuracy. $$\cot x=-3.0649$$.

Estimate the point(s) of intersection. $$l_{1}: 3 x-4 y=7 . \quad l_{2}:-5 x+2 y=11$$

Given that \(x > 0\), compare $$\sqrt{\frac{x}{x+1}} \text { and } \sqrt{\frac{x+1}{x+2}}$$.

Find \(g\) given that \((f g)(x)=c f(x)\).

State whether the function is odd, even, or neither. $$f(x)=x^{3}$$

Let \(x_{0}=2\) and define \(x_{n}=\frac{17+2 x_{n-1}^{3}}{3 x_{n-1}^{2}}\) for \(n=\) 1, 2, 3, 4, .... Find at least five values for \(x_{n}\). Is the set \(S=\left\\{x_{0}, x_{1}, x_{2}, \ldots ., x_{n} \ldots .\right\\}\) bounded above, bounded below, bounded? If so, give a lower bound and/or an upper bound for \(S\). If \(n\) is a large positive integer, what is the approximate value of \(x_{n} ?\)

Find the solutions \(x\) that are in the interval [0.2 \(\pi\) ]. Express your answers in radians and use four decimal place accuracy. $$\sec x=-4.4073$$.

Form the combinations \(f+g . f \quad-g . f\) \(g_{i} f / g\) and specify the domain of combination. $$f(x)=x^{2}-4, \quad g(x)=x+1 / x$$

State whether the function is odd, even, or neither. $$f(x)=x^{2}+1$$

Find the solutions \(x\) that are in the interval [0.2 \(\pi\) ]. Express your answers in radians and use four decimal place accuracy. $$\csc x=10.260$$.

Given that \(a > 0\) and \(b > 0,\) show that if \(a^{2} \leq b^{2}\), then \(a \leq b\).

Estimate the point(s) of intersection. $$l_{1}: 2 x-3 y=5, \quad \text { circle }: x^{2}+y^{2}=4$$

State whether the function is odd, even, or neither. $$g(x)=x(x-1)$$

Write the expression in factored form. \(x^{2}-10 x+25\).

Solve the equation \(f(x)=y_{0}\) for \(x\) in \([0,2 \pi]\) by using a graphing utility. Display the graph of \(f\) and the line \(y=y_{0}\) in one figure; then use the trace function to find the point(s) of intersection. $$f(x)=\sin 3 x ; \quad y_{0}=-1 / \sqrt{2}$$

Show that if \(0 \leq a \leq b\), then \(\sqrt{a} \leq \sqrt{b}\).

Estimate the point(s) of intersection. $$\text { circle }: x^{2}+y^{2}=9, \quad \text { parabola }: y=x^{2}-4 x+5$$

Suppose that \(f\) and \(g\) arc odd functions. What can you conclude about \(f \cdot g ?\)

Form the compositions \(f \circ g\) and \(g \circ f,\) and specify the domain of each of these combinations. $$f(x)=x^{2}-2 x, \quad g(x)=x+1$$

State whether the function is odd, even, or neither. $$g(x)=x\left(x^{2}+1\right)$$

Write the expression in factored form. \(9 x^{2}-4\).

Solve the equation \(f(x)=y_{0}\) for \(x\) in \([0,2 \pi]\) by using a graphing utility. Display the graph of \(f\) and the line \(y=y_{0}\) in one figure; then use the trace function to find the point(s) of intersection. $$f(x)=\cos \frac{1}{2} x ; \quad y_{0}=\frac{3}{4}$$.

The perpendicular bisector of the line scement \(\overline{P Q}\) is the line which is perpendicular to \(\bar{P}(2\) and passes through the midpoint of \(\overline{P Q}\). Find an equation for the perpendicular bisector of the line segment that joins the two points. $$P(-1,3), \quad Q(3,-4)$$

Show that \(|a-b| \leq|a|+|b|\) for all real numbers \(a\) and \(b\).

Suppose that \(f\) and \(g\) arc even functions. What can you conclude about \(f \cdot g ?\)

Form the compositions \(f \circ g\) and \(g \circ f,\) and specify the domain of each of these combinations. $$f(x)=\sqrt{x+1}, \quad g(x)=x^{2}-5$$

State whether the function is odd, even, or neither. $$f(x)=\frac{x^{2}}{1-|x|}$$

Write the expression in factored form. \(8 x^{6}+64\).

Give the domain and range of the function. $$f(x)=|\sin x|$$

Show that ||\(a|-| b|| \leq|a-b|\) for all real numbers \(a\) and \(b\). HINT: Calculate ||\(a|-| b||^{2}\).

Form the compositions \(f \circ g\) and \(g \circ f,\) and specify the domain of each of these combinations. $$f(x)=\sqrt{1-x^{2}}, \quad g(x)=\sin 2 x$$

State whether the function is odd, even, or neither. $$F(x)=x+\frac{1}{x}$$

Write the expression in factored form. \(27 x^{3}-8\).

Give the domain and range of the function. $$g(x)=\sin ^{2} x+\cos ^{2} x$$.

Show that \(|a+b|=|a|+|b|\) iff \(a b \geq 0\).

The points are the vertices of a triangle. State whether the triangle is isosceles (two sides of equal length). a right triangle, both of these, or neither of these. $$P_{0}(-4,3), \quad P_{1}(-4,-1), \quad P_{2}(2,1)$$

For \(x \geq 0 . f\) is defined as follows: $$f(x)=\left\\{\begin{array}{ll}x, & 0 \leq x \leq 1 \\\1, & x>1\end{array}\right.$$ How is \(f\) defined for \(x<0\) if (a) \(f\) is even? (b) \(f\) is odd?

(a) Write an equation in \(x\) and \(y\) for an arbitrary line \(l\) that passes through the origin. (b) Verify that if \(P(a, b)\) lies on \(l\) and \(\alpha\) is a real number, then the point \(Q(\alpha a, \alpha b)\) a)so lies on \(l\) (c) What additional conclusion can you draw if \(\alpha>0 ?\) if \(\alpha<0 ?\)

State whether the function is odd, even, or neither. $$f(x)=\frac{x}{x^{2}-9}$$

Write the expression in factored form. \(4 x^{2}+12 x+9\).

Show that \begin{equation}\text { if } \quad 0 \leq a \leq b, \quad \text { then } \quad \frac{a}{1+a} \leq \frac{b}{1+b}.\end{equation}

The points are the vertices of a triangle. State whether the triangle is isosceles (two sides of equal length). a right triangle, both of these, or neither of these. $$P_{0}(-2,5), \quad P_{1}(1,3) . \quad P_{2}(-1,0)$$