1.2.5What is the symmetry of the function?

Attributes of Even and Odd Functions

1-77.

On two sets of axes, sketch $y = \sin(x)$ on $[–2π, 2π]$ and sketch $y = \cos(x)$ on $[–2π, 2π]$ . Describe their symmetries. Digital Graphing Calculator

1-78.

Decide if the each of the following functions is even, odd, or neither. Use the formal definitions to prove your answers. Then describe the type of symmetry that each function has.

$f(x) = x^2$

$g(x) = 2x^3$

$h(x) = 2 + x^4$

$j(x) = 2 + x^5$

$k(x) = \sin(2x)$

$j(x) = \arctan(x)$

1-79.

Robert wonders about even and odd functions. He wants to know what happens when you combine even functions with odd functions. If $f$ is even and $g$ is odd, determine if the following combinations of $f$ and $g$ are even, odd, or neither. Be sure to explain your decision.

$h(x) = f(x) + g(x)$

$j(x) = f(x) · g(x)$

$k(x) = f(g(x))$

$l(x) = g(f(x))$

$m(x) =| f ( x ) |$

$n(x) =| g ( x ) |$

1-80.

Let $f$ be an even function and $g$ be an odd function, both with domain ${–3, –2, ……, 2, 3}$ and $h(x) = g(f(x))$ .

Complete the table of values below.
$x$
$f(x)$
$g(x)$
$h(x)$
$−3$
$1$
$1$
$−2$
$2$
$2$
$−1$
$1$
$1$
$0$
$0$
$0$
$1$
$2$
$3$
What can you conclude about $h$ ?
Determine the following values. If it is impossible, justify why.
1. $h(4)$
2. $g^{−1}(2)$
3. $f(g(−1))$

1-81.

When even and odd functions have asymptotes, interesting things happen. Consider the following situations.

If $f$ is an odd function such that $y = k$ is a horizontal asymptote, which of the following must be true?
1. $y = -k$ is a horizontal asymptote.
2. $x = k$ is a vertical asymptote.
3. $x = -k$ is a vertical asymptote.
Allie conjectures that an odd function cannot have just one unique horizontal asymptote; it either has two opposite horizontal asymptotes, or none. Is she correct? If so, explain. If not, give a counterexample.

1-82.

Are the following functions even, odd, or neither? Homework Help ✎

$y = x^{1/3}$
$y = −x^2 + 4$
$y = x^3 + x^2 + 1$

1-83.

If $f(x) = x^2 + 5x$ and $g(x) = x + 3$ , evaluate each of the following expressions. Homework Help ✎

$f(−2)$
$g(−2)$
$f(g(−2))$
$g(f(−2))$
$f(f(−2))$
$g(g(−2))$

1-84.

Examine the graph of $f(x) = 0.5x - 2$ at right. Homework Help ✎

Calculate the area of the shaded region using geometry. (Recall that area below the $x$ -axis is considered negative.)
What is the value of $k$ if the area under the curve for $0 ≤ x ≤ k$ is $10$ ? How did you obtain your solution?

Increasing line, passing through the points (0, comma negative 2) & (4, comma 0), with the triangular region right of the y axis, above the line, & below the x axis, shaded, & the region below the line & above the x axis, left of x = 7, shaded.

1-85.

Graph $f(x) = 2x +\large\frac { 1 } { x }$ . What is the end-behavior function? 1-85 HW eTool Homework Help ✎

1-86.

A flag is defined by the region between the $x$ -axis and:

$f ( x ) = \left\{ \begin{array} { c c c } { - x + 5 } & { \text { for } } & { 1 \leq x \leq 3 } \\ { 2 } & { \text { for } } & { 3 < x \leq 6 } \end{array} \right.$

Calculate the volume generated when the flag is rotated about the $x$ -axis. 1-86 HW eTool Homework Help ✎

1-87.

Review the directions for writing approach statements in the Math Notes box in Lesson 1.2.3. Then write a complete set of approach statements for $y = 3^{−x} + 1$ . 1-87 HW eTool Homework Help ✎

1-88.

If $y = \large\frac { x } { x + 1 }$ , approximate the area under the curve for $0 ≤ x ≤ 4$ using eight left endpoint rectangles of equal width. 1-88 HW eTool Homework Help ✎

1-89.

If $f(x) = x^2 + 5$ and $g(x) = x + 3$ , write and simplify expressions for the function operations given below. Homework Help ✎

$f(g(x))$
$g(f(x))$
$f^{ −1}(−6)$
$g^{−1}(−6)$

1-90.

Compute without a calculator Without a calculator, sketch each graph, showing roots, holes, and asymptotes. Then, state the domain in parts (a) and (b) using interval notation and the domain in parts (c) and (d) using set notation. 1-90 HW eTool Homework Help ✎

$y =\large\frac { x ^ { 2 } - 4 } { x ^ { 3 } + 3 x ^ { 2 } - 10 x }$
$y =\large\frac { 9 - x ^ { 2 } } { 2 x + 6 }$
$y =\large\frac { x ^ { 2 } - 9 x - 18 } { x ^ { 2 } + 3 x - 18 }$
$y =\large\frac { x ^ { 2 } - 6 x + 9 } { 15 - 5 x }$

1-91.

The domain of a function $f$ is $x > 0$ . The range of $f$ is $−2 < y ≤ 5$ . Homework Help ✎

Sketch a possible graph of $f$ .
What are the domain and range of $y = f(x - 2) +1$ ?
What are the domain and range of $f^{ -1}$ ?

$x$	$f(x)$	$g(x)$	$h(x)$
$−3$	$1$	$1$
$−2$	$2$	$2$
$−1$	$1$	$1$
$0$	$0$	$0$
$1$
$2$
$3$

Calculus Third Edition

Even and Odd Functions