(b) f(x) = −x
f(−x) = −(−x) = x
−f(x) = −(−x) = x
So this means that the function is odd.
(c) f(x) = (x − 1)2
This function is neither odd nor even. To see why, observe that
f(−x) = (−x − 1)2 = ((−1)(x + 1))2 = (−1)2(x + 1)2 = (x + 1)2.
f(−x) = (x + 1)2.
This is clearly not the same as
f(x) = (x − 1)2
(plug in 2 for x to see that they are different) so the function is not even.
Similarly, f(−x) is not the same as
−f(x) = −(x − 1)2
(plug in 0 for x to see that they are different) so the function is not odd.
Therefore, it is neither.
8. Solve each inequality by graphing an appropriate function, and state the
set using interval notation. Here are the inequalities.
(1) (x − 1)2 − 9 < 0
(2) (x − 1)2 − 9 0
The appropriate function is f(x) = (x−1)2 −9 I won’t sketch the graph, but I
describe it in sufficient detail to enable you to graph it. The graph is a
that opens up and has its vertex at (1, -9). Its y-intercept is f(0) = 1 − 9 =
To find its x- intercepts , solve the equation
(x − 1)2 = 9
x − 1 = ±3
so x = −2 or x = 4.
13. Show that the function given by y = |x + 2| is not one-to-one.
We need to show that there are two different values x = a and x = b such that
|a + 2| = |b + 2|.
We can think of such values by contemplating the meaning of an absolute value.
Now |x+2| stands for the distance of x from -2. We need to find different
that are the same distance away from -2. To do this, imagine -2 on the number line , and take one step to the right arriving at -1, or one step to the left,
at -3. Note that
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