0. Functions

Assume f is an odd function and that both f and g are one-to-one. Use the (incomplete) graph of f and the graph of g to find the following function values. <IMAGE>

f⁻¹ (10)

Assume f is an odd function and that both f and g are one-to-one. Use the (incomplete) graph of f and the graph of g to find the following function values. <IMAGE>

f^-1(1 + f(-3))

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.

70. y= x³/(x²+1), -1 ≤ x ≤ 1, x_0=1/2

[Technology Exercise]

a. Graph the functions f(x) = 3/(x − 1) and g(x) = 2/(x + 1) together to identify the values of x for which

3/(x − 1) < 2/(x + 1)

b. Confirm your findings in part (a) algebraically.

Demand function Sales records indicate that if Blu-ray players are priced at \$250, then a large store sells an average of 12 units per day. If they are priced at \$200, then the store sells an average of 15 units per day. Find and graph the linear demand function for Blu-ray sales. For what prices is the demand function defined?

Even and Odd Functions

In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.

h(t) = |t³|

Find the domain of the rational function. Then, write it in lowest terms.

$f\(\left\)(x\(\right\))=\(\frac{6x^5}{2x^2-8}\)$

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

$f\(\left\)(x\(\right\))=10^{-2x}$

Find the inverse function (on the given interval, if specified) and graph both $f$ and $f^{-1}$ on the same set of axes. Check your work by looking for the required symmetry in the graphs.

$f\(\left\)(x\(\right\))=8-4x$

Solving trigonometric equations Solve the following equations.

sin Θ cos Θ = 0, 0 ≤ Θ < 2π

Theory and Examples

The variables r and s are inversely proportional, and r = 6 when s = 4. Determine s when r = 10.

In the graph shown, identify the y–intercept & slope. Write the equation of this line in Slope-Intercept form.

Finding inverses Find the inverse function.

ƒ(x) = 3x - 4

Find the linear function whose graph passes through the point (3, 2) and is parallel to the line $y=3x+8$ .

Inverse of composite functions

b. Let g(x) = x² + 1 and h(x) = √x. 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