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Cumulative Review Exercises Chapters P–4

Find the x- and y-intercepts of the graph of the equation $x^{2} + {(y - 1)}^{2} = 1 .$ $x^{2} + {(y - 1)}^{2} = 1 .$ Check for symmetry with respect to both axes and the origin.
Graph the equation $y = x^{3} .$ $y = x^{3} .$
Find the center and radius of the circle $x^{2} + 2 x + y^{2} - 4 y - 20 = 0 .$ $x^{2} + 2 x + y^{2} - 4 y - 20 = 0 .$ Graph the circle.
Write the slope–intercept form of the equation of the line that passes through $(- 2, 4)$ $(- 2, 4)$ and is perpendicular to the line $2 x + 3 y = 17 .$ $2 x + 3 y = 17 .$
Find the domain of $f (x) = \log \frac{x - 2}{x + 3} .$ $f (x) = \log \frac{x - 2}{x + 3} .$
Evaluate the function $f (x) = {\begin{array}{l} 2 x - 3, & x < 1 \\ 2 x^{2} + 1, & x \geq 1 \end{array}$ $f (x) = {\begin{array}{l} 2 x - 3, & x < 1 \\ 2 x^{2} + 1, & x \geq 1 \end{array}$ at each value specified.
1. $f (- 2)$ $f (- 2)$
2. f(0)
3. f(3).
Identify the basic function and use transformations to sketch the graph of the function $f (x) = 3 \sqrt{x + 1} - 2 .$ $f (x) = 3 \sqrt{x + 1} - 2 .$
Let $f (x) = 3 \sqrt{x}$ $f (x) = 3 \sqrt{x}$ and $g (x) = - \frac{1}{x} .$ $g (x) = - \frac{1}{x} .$

Find and write the domain of $(f \circ g) (x)$ $(f \circ g) (x)$ in interval notation.
Determine whether the function has an inverse function; if so, find the inverse function.
1. $f (x) = 2 x^{3} + 1$ $f (x) = 2 x^{3} + 1$
2. $f (x) = | x |$ $f (x) = | x |$
3. $f (x) = \ln x .$ $f (x) = \ln x .$
Divide
1. $\frac{2 x^{3} + 3 x + 1}{x^{2} + 1}$ $\frac{2 x^{3} + 3 x + 1}{x^{2} + 1}$ using long division.
2. $\frac{2 x^{4} + 7 x^{3} - 2 x + 3}{x + 3}$ $\frac{2 x^{4} + 7 x^{3} - 2 x + 3}{x + 3}$ using synthetic division.
Find a polynomial function of least degree with integer coefficients that has the given zeros.
1. $1, 2, - 3$ $1, 2, - 3$
2. $1, - 1, i, - i$ $1, - 1, i, - i$
Use rules of logarithms to rewrite the following expressions in expanded form.
1. $\log_{2} 5 x^{3}$ $\log_{2} 5 x^{3}$
2. $\log_{a} \sqrt[3]{\frac{x y^{2}}{z}}$ $\log_{a} \sqrt[3]{\frac{x y^{2}}{z}}$
3. $\ln \frac{3 \sqrt{x}}{5 y}$ $\ln \frac{3 \sqrt{x}}{5 y}$
Use rules of logarithms to rewrite the following expressions in condensed form.
1. $\frac{1}{2} (\log x + \log y)$ $\frac{1}{2} (\log x + \log y)$
2. $3 \ln x - 2 \ln y$ $3 \ln x - 2 \ln y$
Use a calculator to evaluate the expression. Round your answer to three decimal places.
1. $\log_{9} 17$ $\log_{9} 17$
2. $\log_{3} 25$ $\log_{3} 25$
3. $\log_{1 / 2} 0.3$ $\log_{1 / 2} 0.3$
Solve each equation.
1. $\log_{6} (x + 3) + \log_{6} (x - 2) = 1$ $\log_{6} (x + 3) + \log_{6} (x - 2) = 1$
2. $5^{x^{2} - 4 x + 5} = 25$ $5^{x^{2} - 4 x + 5} = 25$
3. ${3.1}^{x - 1} = 23$ ${3.1}^{x - 1} = 23$
Find all possible rational zeros of

$f (x) = x^{4} - x^{3} - 2 x^{2} - 2 x + 4 .$ $f (x) = x^{4} - x^{3} - 2 x^{2} - 2 x + 4 .$

Also find upper and lower bounds for the zeros of f.
Let $f (x) = {(x - 1)}^{3} {(x + 2)}^{2} (x - 3) .$ $f (x) = {(x - 1)}^{3} {(x + 2)}^{2} (x - 3) .$
1. Find the zeros of f along with their multiplicities.
2. What is the behavior of f near each zero?
3. What is the end behavior of f?
4. Sketch the graph of f.
Let $f (x) = \frac{(x - 1) (x + 2)}{(x - 3) (x + 4)} .$ $f (x) = \frac{(x - 1) (x + 2)}{(x - 3) (x + 4)} .$
1. Find the vertical asymptotes of f (if any).
2. Find the horizontal asymptotes of f (if any).
3. Find the intervals over which $f (x) - 1 > 0$ $f (x) - 1 > 0$ or $f (x) - 1 < 0 .$ $f (x) - 1 < 0 .$
4. Sketch the graph of f.

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Table of Contents for Cumulative Review Exercises Chapters P&#8211;4

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Table of Contents for
Cumulative Review Exercises Chapters P–4