Other functions

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Other functions

This kind of function is one in which the variable (i.e. the x) is an index or exponent (i.e. is the power).

For example:

3x or 52-3x + 9 or e-x

For exponential functions in the form ax these are the important facts:

  • f(x) > 0 for all real values of x.
  • f(x) = 1 when x = 0.
  • f(x) increases at an accelerating rate as x increases.
  • f(x) becomes smaller as x decreases.
  • f(x) approaches zero as x approaches negative infinity. You may see this written as: as x → −∞, f(x) → 0.

Example:

The graph of f(x) ≡ 3x can be drawn as:

Example

The curve gets very close to the negative x-axis but never touches it. When this happens, we say that the x-axis is an asymptote to the curve.

An alternative way of stating that f(x) approaches a limit of zero as x approaches negative infinity can be written as :

Limit

For any value of k > 1, the exponential function of the form kx is similar to the graph f(x) ≡ 3x drawn above.

(For further information on logarithmic functions see the Algebra Learn-It.)

In order to sketch to logarithmic function you need to learn a few rules.

1. If you have a logarithmic function f(x) ≡ log ax, then the function is undefined (it does not exist) for x ≤ 0
2. f(x) > 0 for x > 1 and as x → ∞, f(x) → ∞.
3. f(x) = 0 for x = 1 (ao = 1 therefore log a1 = 0)
4. f(x) < 0 for 0 < x < 1 as x decreases to zero from a positive side.

The sketch of f(x) ≡ log ax and any function of this form will have a similar shape to this graph:

Graph of log

Note: This is the inverse of f(x) = ax and hence is a reflection of the previous graph in the line y = x.

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