MATH 150 PRELIMINARY NOTES 4
INVERSE FUNCTIONS AND LOGARITHMS
INVERSE FUNCTIONS
Let us think back to the days of algebra and to the time we
spent learning about functions, recall that a function is a rule
(or equation) that assigns to every value in the domain a
single value in the range. Also recall that the easiest way to
determine if a relation is a function is to use the vertical line
test.
VERTICAL LINE TEST: Draw any vertical line. If this line
intersects the graph in only one place, then the relation is a
function. (See figure 1)
figure 1
After we learned about functions, we then moved on to the
concept of an inverse function. To determine if a function's
inverse is a function, we have to determine if the original
function is one-to-one. A function is one-to-one if for every
value in the range, there is exactly one value in the
domain. The easiest way to determine if a function is one-toone is to use the horizontal line test.
HORIZONTAL LINE TEST: Draw any horizontal line. If
this line intersects the graph at only one place, then the
function is one-to-one. (See figure 2)
FACT: If a function is one-to-one, then it's inverse is a
function.
figure 2
FACT: A function and it's inverse are symmetric about the
line y = x.(See figure 3)
Now let us work some examples of determining if a function is
one-to-one and then find its inverse.
figure 3
EXAMPLE 1:
Determine if f (x) = (x - 1) 2 is one-to-one. If it is, then find its
inverse.
SOLUTION: The easiest way to determine if this function is
one-to-one is to look at the graph of the function and use the
horizontal line text on it. (See figure 4)
It is not one-to-one, so it's inverse is not a function.
figure 4
EXAMPLE 2:
Determine if f (x) = x 2 + 2, x ≥ 0 is one-to-one. If it is, then
find its inverse.
SOLUTION: The easiest way to determine if this function is
one-to-one is to look at the graph of the function and use the
horizontal line text on it. (See figure 5)
It is one-to-one because we are only considering the right
branch of the parabola.
figure 5
Now to determine the inverse. First we will interchange x with y.
x = y2 + 2
Now we will solve for y in terms of x.
Notice that we used the positive root of the square root. The reason for this comes from the condition, x ≥ 0, that was imposed on the original function.
EXAMPLE 3:
Determine if
is one-to-one. If it is, then find its inverse.
SOLUTION: The easiest way to determine if this function is one-toone is to look at the graph of the function and use the horizontal line
text on it. (See figure 6)
figure 6
It is one-to-one, so we must find it's inverse. We will first interchange x with y, and then we will solve for y in terms of x.
Now that we have spent some time on inverse functions, let us turn our attention to the inverse of an exponential function. What is the inverse of the
exponential function? It is a logarithmic function, and here is the definition.
DEFINITION:
The base a logarithm function y = log a x is the inverse of the base a exponential function y = a x (a > 0, a ≠ 1).
There are two basic logarithmic functions: the common log (log base 10) and the natural log (log base e). The common log function is y = log x, and the
natural log function is y = ln x. All logarithmic functions have a starting domain of (0, ∞ ) and a range of (-∞ , ∞ ).
For log base a: a logax = x and log a a x = x, a > 0, a ≠ 1, x > 0.
FACT:
For log base e: e ln x = x and ln e x = x, x > 0.
PROPERTIES OF LOGARITHMS
For any real numbers x > 0 and y > 0,
1.
log a xy = log a x + log a y
2.
3.
log a x y = y log a x
Now we will solve some logarithmic and exponential equations.
EXAMPLE 4:
Solve e x + e - x = 3 for x.
SOLUTION:
e 2 x + 1 = 3e x → e 2 x - 3e x + 1 = 0
Now let p = e x.
e 2 x - 3e x + 1 = 0 → p 2 - 3p + 1 = 0
This quadratic does not factor, so we will have to use the quadratic formula.
Now set e x equal to each of the solutions and solve for x.
EXAMPLE 5:
Solve for y: ln y = 2t + 4
SOLUTION: To solve this equation for y, use the fact that e ln y = y, so write both sides as e to a power.
ln y = 2t + 4 → e ln y = e 2 t + 4 → y = e 2 t + 4
This concludes the review of inverse and logarithmic functions. Work through the examples that I have provided in this set of notes, and if you have any
questions, please feel free to contact me.
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