Technical Mathematics, Sixth Edition

17.5. Inverse Variation

▪ Exploration:

The two functions

y = x and y = 1/x

look somewhat alike. (a) Can you predict, for each, what happens to y as x gets larger? (b) What happens to y for each as x gets very large? (c) What happens to y for each as x gets very small?

Try this. Graph the two functions in the same viewing window, for x = 0 to 3. Does your graph bear out your predictions?

For the bar in tension, Fig. 17-17, the stress σ is equal to the applied force P divided by the cross-sectional area a of the bar, or . What happens mathematically to the stress as a increases? As a gets very small? Does the mathematics agree with what you know about the stress in a bar?

Figure 17.17. A bar in tension.

When we say that "y varies inversely as x" or that "y is inversely proportional to x," we mean that x and y are related by the following equation, where, as before, k is a constant of proportionality:

The equation y = k/x can also be written as

y = kx⁻¹

that is, a power function with a negative exponent. Another form is obtained by multiplying both sides of y = k/x by x, getting

xy = k

Each of these three forms indicate inverse variation. Inverse variation problems are solved by the ...