Discussion – Wednesday, March Day
Algebra Review
1. Find the value of x that satisfies the following equations:
x−1
=6
(a) x+1
6
(b) 1−x
2 = 14
x
(c) 2√ = 16
x+1+1=4
(d)
2. Complete the square for each expression.
(a) x2 − 4x + 2
(b) x2 + 3x + 9
(c) 2x2 − 12x + 5
Function Inverses
3. Let f (x) = ex and g(x) = lnx.
(a) Determine the domain and range of f (x) and g(x).
(b) Determine where f −1 (x) and g −1 (x) exists.
(c) Graph f (g(x)) and g(f (x)).
(d) Explain how the relationship between the domain of g(x) and range of g −1 (x) affected
your answer in part c.
3. For each polynomial, find a domain on which f (x) has an inverse, yet the range is
unchanged.
(a) (x − 1)2 (x + 4)2
(b) (x − 1)(x − 3)
4. Show that no even degree polynomial, p(x), is not one-to-one by showing the following
steps:
(a) Find the end behavior
(b) Let m ∈ R be a min or max. Explain why we know there is a y-value in between m
and the lim p(x); m and lim p(x)
x→∞
(c)
x→−∞
Explain how this fact yields the desired .
5. Is it possible for a function to be its own inverse? If not explain why, if yes then give an
example.
6. Give an example of function so that the domain on which f −1 (x) exists is different from
the domain of f −1 (x)
7. Let f (x) = x5 + x3 + 12x + 6. Find f −1 (6) and f −1 (20) (Hint: f (x) is strictly increaing.
How does this help?)
Logarithms
8. Show how we can derive the law log(ac ) = c log(a) from the law log(ab) = log(a) + log(b)
9. From the previous two laws, derive the law log ab = log(a) − log(b).
2 4
10. Fully expand the expression log acdb3
11. Fully Simplify the following:
(a) 2 log4 8
log 9
(b) log
3
2+log 24.5
(c) log
log 28−log 4