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Naddik [55]
3 years ago
7

How does the graph show that the rate of change is constant

Mathematics
1 answer:
ElenaW [278]3 years ago
7 0

A graph can show that a rate of change is constant as long as the graph is of a straight line.

Any curvature in the line suggests a rate of change that is variable, but otherwise it is constant.

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What is the domain of F(x) = In(x)?
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Answer as an inequality:  x > 0

Answer in interval notation:  (0, \infty)

Answer in words: Set of positive real numbers

All three represent the same idea, but in different forms.

======================================================

Explanation:

Any log is the inverse of an exponential equation. Consider a general base b such that f(x) = b^x. The inverse of this is f^{-1}(x) = \log_b(x)

For the exponential b^x, we cannot have b^x = 0. We can get closer to it, but we can't actually get there. The horizontal asymptote is y = 0.

Because of this, \log_b(x) has a vertical asymptote x = 0 (recall that x and y swap, so the asymptotes swap as well). This means we can get closer and closer to x = 0 from the positive side, but never reach x = 0 itself.

The domain of \log_b(x) is x > 0 which in interval notation would be (0, \infty). This is the interval from 0 to infinity, excluding both endpoints.

------------------------

The natural log function Ln(x) is a special type of log function where the base is b = e = 2.718 approximately.

So,

\log_e(x) = \text{Ln}(x)

allowing all of what was discussed in the previous section to apply to this Ln(x) function as well.

------------------------

In short, the domain is the set of positive real numbers. We can't have x be 0 or negative.

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