0. Functions

Theory and Examples

In Exercises 69 and 70, match each equation with its graph. Do not use a graphing device, and give reasons for your answer.

a. y = x⁴

b. y = x⁷

c. y = x¹⁰

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In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

b. Solve the equation y=f(x) for x as a function of y, and name the resulting inverse function g.

72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2

Draining a tank (Torricelli’s law) A cylindrical tank with a cross-sectional area of $10$ m² is filled to a depth of $25$ m with water. At $t=0$ s, a drain in the bottom of the tank with an area of $1$ m²  is opened, allowing water to flow out of the tank. The depth of water in the tank (in meters) at time $t\(\geq{0}\)$ is $d\(\left\)(t\(\right\))=\(\left\)(5-0.22t\(\right\))^2$.

a. Check that $d\(\left\)(0\(\right\))=25$, as specified.

Theory and Examples

The accompanying figure shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 2 units long.

a. Express the y-coordinate of P in terms of x. (You might start by writing an equation for the line AB.)

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Determine if the function is an exponential function.  

If so, identify the power & base, then evaluate for $x=4$ .

$f\(\left\)(x\(\right\))=\(\left\)(\(\frac\)12\(\right\))^{x}$

Identify the ordered pair of the vertex of the parabola. State whether it is a minimum or maximum.

{Use of Tech} Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h= ƒ(t) = 64t - 16t²  where t is measured in seconds after the hit.

c. Find the inverse function that gives the time t at which the ball is at height h as the ball travels downward. Express your answer in the form t = ƒ⁻¹ (h)

Evaluate cos⁻¹(cos(5π/4)).

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

If $f\left(x\right)=x^2+1$ , then $f^{-1}\left(x\right)=\frac{1}{x^2+1}$.

Find the inverse $f^{-1}\(\left\)(x\(\right\))$ of each function (on the given interval, if specified).

$f\(\left\)(x\(\right\))=\(\frac{x}{x-2}\)$, for $x\(\gt{2}\)$

A pen in the shape of an isosceles right triangle with legs of length x ft and hypotenuse of length h ft is to be built. If fencing costs \$5/ft for the legs and \$10/ft for the hypotenuse, write the total cost C of construction as a function of h.

Yeast growth Consider a colony of yeast cells that has the shape of a cylinder. As the number of yeast cells increases, the cross-sectional area A (in mm²) of the colony increases but the height of the colony remains constant. If the colony starts from a single cell, the number of yeast cells (in millions) is approximated by the linear function N(A) - CₛA, where the constant Cₛ is known as the cell-surface coefficient. Use the given information to determine the cell-surface coefficient for each of the following colonies of yeast cells, and find the number of yeast cells in the colony when the cross-sectional area A reaches 150 mm². (Source: Letters in Applied Microbiology, 594, 59, 2014)

The scientific name of baker’s or brewer’s yeast (used in making bread, wine, and beer) is Saccharomyces cerevisiae. When the cross-sectional area of a colony of this yeast reaches 100 mm², there are 571 million yeast cells.

Determine if the function is an exponential function.  

If so, identify the power & base, then evaluate for $x=4$ .

$f\(\left\)(x\(\right\))=3\(\left\)(1.5\(\right\))^{x}$

Draining a tank (Torricelli’s law) A cylindrical tank with a cross-sectional area of $10$ m² is filled to a depth of $25$ m with water. At $t=0$ s, a drain in the bottom of the tank with an area of $1$ m²  is opened, allowing water to flow out of the tank. The depth of water in the tank (in meters) at time $t\(\geq{0}\)$ is $d\(\left\)(t\(\right\))=\(\left\)(5-0.22t\(\right\))^2$.

c. What is an appropriate domain for $d$?

Inverse of composite functions

a. Let g(x) = 2x + 3 and h(x) = x³. Consider the composite function ƒ(x) = g(h(x)). Find ƒ⁻¹ directly and then express it in terms of g⁻¹ and h⁻¹