Name: SOLUTION
Math 1A Quiz #1
Section:
Each problem is worth 3 points. Show all your work to receive full credit.
1. Evaluate and simplify the difference quotient ((f (a + h) − f (a))/h for the function f (x) = x3 . State when
your expression is valid.
We plug our function f (x) = x3 into the expression to obtain
(a + h)3 − a3
h
=
=
=
(a3 + 3a2 h + 3ah2 + h3 ) − a3
h
3a2 h + 3ah2 + h3
h
3a2 + 3ah + h2
whenever h 6= 0.
Does the function f (x) = 2x3 + 3 have an inverse? If so, explain why and find it. If not, explain.
Recall that a function is invertible exactly if it is 1 to 1.
The graph of f (x) is the graph of x3 stretched vertically and then shifted up by 3. Since x3 is 1 to 1, f (x) is
as well.
To find the inverse we set y = f (x) = 2x3 + 3 and solve for x in terms of y. Thus
2x3
=
x3
=
x =
y−3
y−3
r2
3 y − 3
.
2
Notice that each step in this calculation is reversible, which gives us another way to see that f is 1 to 1.
Finally, to check our answer,
!3
r
y
−
3
y−3
3
f (f −1 (y)) = 2
+3=3·
+3=y
2
2
r
r
3
3
√
3 (2x + 3) − 3
3
3 2x
−1
f (f (x)) =
=
= x3 = x.
2
2
This gives us one more way to argue that f is invertible.
2. Find the exact value of the expression log8 (40) − log8 (3log3 (5) ).
Using the standard rules for manipulation of logarithm, we obtain
40
log8 (40) − log8 (5) = log8
= log8 (8) = 1.
5