91Ó°ÊÓ

Determine whether the function has an inverse function. If it does, then find the inverse function. $$f(x)=x^{4}$$

Find a mathematical model that represents the statement. (Determine the constant of proportionality.) \(y\) is inversely proportional to \(x .(y=7 \text { when } x=4 .)\)

The number \(N\) of bacteria in a refrigerated food is given by $$N(T)=10 T^{2}-20 T+600, \quad 2 \leq T \leq 20$$ where \(T\) is the temperature of the food in degrees Celsius. When the food is removed from refrigeration, the temperature of the food is given by $$T(t)=3 t+2, \quad 0 \leq t \leq 6$$ where \(t\) is the time in hours. (a) Find the composition \((N \circ T)(t)\) and interpret its meaning in context. (b) Find the bacteria count after 0.5 hour. (c) Find the time when the bacteria count reaches 1500 .

Find a mathematical model that represents the statement. (Determine the constant of proportionality.) \(P\) varies directly as \(x\) and inversely as the square of \(y\) \(\left(P=\frac{28}{3} \text { when } x=42 \text { and } y=9 .\right)\)

The simple interest on an investment is directly proportional to the amount of the investment. An investment of \(\$ 6500\) will earn \(\$ 211.25\) after 1 year. Find a mathematical model that gives the interest \(I\) after 1 year in terms of the amount invested \(P\).

Prove that the product of two odd functions is an even function, and that the product of two even functions is an even function.

Use Hooke's Law for springs, which states that the distance a spring is stretched (or compressed) varies directly as the force on the spring. A force of 220 newtons stretches a spring 0.12 meter. What force is required to stretch the spring 0.16 meter?

Geometry A right triangle is formed in the first quadrant by the \(x\) - and \(y\) -axes and a line through the point (2,1) (see figure). Write the area \(A\) of the triangle as a function of \(x,\) and determine the domain of the function.

Use Hooke's Law for springs, which states that the distance a spring is stretched (or compressed) varies directly as the force on the spring. A force of 265 newtons stretches a spring 0.15 meter. (a) What force is required to stretch the spring 0.1 meter? (b) How far will a force of 90 newtons stretch the spring?

The diameter of the largest particle that can be moved by a stream varies approximately directly as the square of the velocity of the stream. A stream with a velocity of \(\frac{1}{4}\) mile per hour can move coarse sand particles about 0.02 inch in diameter. Approximate the velocity required to carry particles 0.12 inch in diameter.