Chapter 1.5 Functions and Logarithms
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Transcript of Chapter 1.5 Functions and Logarithms
Chapter 1.5Functions and Logarithms
One-to-One Function A function f(x) is one-to-one on a domain D (x-axis) if f(a) ≠ f(b) whenever
a≠b
Use the Horizontal line test
The graph of a one-to-one function y = f(x) can intersect any horizontal line at most once. If a horizontal intersects a graph more than once, the function is not one.
If a function is one-to-one it has an inverse
Horizontal Line Test Examples X3 x2
Finding the inverse function If a function is one-to-one it has an inverse
Writing f-1 as a Function of x
1) Solve the equation y = f(x) for x in terms of y.
2) Interchange x and y. The resulting formula will be y = f-1 (x)
Inverse Examples Show that the function y = f(x) = -2x +4 is one-to-one and find its inverse
Every horizontal line intersects the graph of f exactly once, so f is one-to-one and has an inverse
Step 1: Solve for x in terms of Y:
Y = -2x + 4
X= -(1/2)y +2
Step 2: Interchange x and y: y = -(1/2)x + 2
Logarithmic Functions The base a logarithm function y = logax is the inverse of the base a
exponential function y = ax
The domain of logax is (0,∞). The range of logax is (-,∞, ,∞)
Important Log Functions Two very important logs for conversions and our calculators are:
The common log function Log10x = logx
The natural log Logex = lnx
Properties of Logarithms Inverse properties for ax and logax
1) Base a: aloga(x) = x, logaax = x, a > 1, x > 0
2) Base e: elnx = x, lnex = x
Examples: Solve for x 1) lnx = 3y + 5
2) e2x = 10
Properties of Logarithms For any real number x > 0 and y > 0
1) Product Rule: logaxy = logax + logay
2) Quotient Rule: loga(x/y) = logax – logay
3) Power Rule: logaxy = ylogax
4) Change of Base Formula: logax = (lnx)/(lna)
Investment Sarah invests $1000 in an account that earns 5.25% interest compounded annually.
How long will it take the account to reach $2500?
P(1+(r/c))ct=A
1000(1.0525)t = 2500
(1.0525)t = 2.5
Ln(1.0525)t = ln2.5
Tln1.0525 = ln2.5
T = (ln2.5)/(ln1.0525) = 17.9
Homework Quick Review: pg 43, # 1, 3, 7, 9
Exercises: pg 44, # 1, 2, 3, 6, 7, 8, 10, 33, 34, 37, 39, 40, 47, 48