Page 189 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes 6.3.6 Finding Inverses

Example: First consider a simple example f(x) = 3x + 2.

The graph of f is a line with slope 3, so it passes the horizontal line test and does have an inverse.
There are two steps required to evaluate f at a number x. First we multiply x by 3, then we
add 2.

Thinking of the inverse function as undoing what f did, we must undo these steps in reverse
order.
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The steps required to evaluate f are to first undo the adding of 2 by subtracting 2. Then we undo
multiplication by 3 by dividing by 3.
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Therefore, f (x) = (x - 2)/3.
Steps for finding the inverse of a function f.

1. Replace f(x) by y in the equation describing the function.
2. Interchange x and y. In other words, replace every x by a y and vice-versa.
3. Solve for y.
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4. Replace y by f (x).
6.4 Logarithmic Function

The logarithmic function is defined as the inverse of the exponential function.
Parameters included in the definition of the logarithmic function may be changed, using sliders,
to investigate its properties. The continuous (small increments) changes of these parameters
help in gaining a deep understanding of logarithmic functions. The function to be explored has
the form

f(x) = a*logB[ b (x+c) ] + d
a, b, c and d are coefficients and B is the base of the logarithm.
For B > 0 and B not equal to 1, y = Log B is equivalent to x = B .
x
y

Notes The logarithm to the base e is written ln(x).

Example:
1. f(x) = log2x

2. g(x) = log4x
3. h(x) = log0.5x
Consider the function given below:
y = e x …(1)

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