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

$f\left(x\right)=\ln\left(3x+1\right)$

Where do inverses exist? Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse.

ƒ(x) = |2x + 1|

Convert the following expressions to the indicated base.

$a^{\frac{1}{\ln a}}$ using basa e, for $a>0$ and $a\ne 1$

For a certain constant a>1, ln a≈3.8067 . Find approximate values of  log₂ a and logₐ 2 using the fact that ln 2≈0.6931.

Suppose ƒ is an even function with ƒ(2) = 2 and g is an odd function with g(2) = -2. Evaluate ƒ(-2) , ƒ(g(2)), and g(ƒ(-2))

Simplify the difference quotient ƒ(x+h)-ƒ(x)/h

ƒ(x) = (x)/(x+1)

Intersection problems Find the following points of intersection.

The point(s) of intersection of the parabolas y= x² and y= -x² + 8x

Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-a) for the following functions.

ƒ(x) = 4 - 4x + x²

Simplify the difference quotients ƒ(x+h) - ƒ(x) / h and ƒ(x) - ƒ(a) / (x-a) by rationalizing the numerator.

ƒ(x) = - (3/√x)

More composite functions Let ƒ(x) = | x | , g(x)= x² - 4 , F(x) = √x , G(x) = (1)/(x-2) Determine the following composite functions and give their domains.

G o G

Use the graph of ƒ to find ƒ⁻¹ (2),ƒ⁻¹ (9), and ƒ⁻¹ (12) <IMAGE>

Missing piece Let g(x) = x² + 3 Find a function ƒ  that produces the given composition.

(g o ƒ ) (x) = x⁴ + 3

Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-a) for the following functions.

ƒ(x) = (1/x) - x²

Finding all inverses Find all the inverses associated with the following functions, and state their domains.

ƒ(x) = (x + 1)³

Solve each equation.

$48=6e^{4k}$

Finding all inverses Find all the inverses associated with the following functions, and state their domains.

ƒ(x) = 2 / ( x² + 2)

Properties of logarithms Assume logbx = 0.36, logby= 0.56 and logbz = 0.83 . Evaluate the following expressions.

logb x/y

Simplify the difference quotients ƒ(x+h) - ƒ(x) / h and ƒ(x) - ƒ(a) / (x-a) by rationalizing the numerator.

ƒ(x) = √(1-2x)

Working with composite functions

Find possible choices for outer and inner functions ƒ and g such that the given function h equals ƒ o g.

h(x) = (2) / ( x⁶ + x² + 1)²

Find the inverse of the function ƒ(x) = 2x. Verify that ƒ(ƒ⁻¹(x)) = x and ƒ⁻¹(ƒ(x)) = x .

Sketch the graph of the inverse of ƒ. <IMAGE>

Solving equations Solve the following equations.

5(ˣ³) = 29

Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse.

{Use of Tech} $f\left(x\right)=\frac{1}{x^2}$

Simplify the difference quotient ƒ(x+h)-ƒ(x)/h

ƒ(x) = 2x² -3x +1

Properties of logarithms Assume logbx = 0.36, logby= 0.56 and logbz = 0.83 . Evaluate the following expressions.

logb (√x) / (³√z)

Solving equations Solve each equation.

√2 sin 3Θ + 1 = 2, 0 ≤ Θ ≤ π

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

$f\left(x\right)=e^{2x+6}$

Working with composite functions Find possible choices for outer and inner functions ƒ and g such that the given function  h equals ƒ o g .

h(x) = √ (x⁴ + 2 )

Inverse sines and cosines Evaluate or simplify the following expressions without using a calculator.

cos⁻¹ √3/2

Graphing functions Sketch a graph of each function.

g(x) = { 4-2x if x ≤ 1 , (x-1)² + 2 if x > 1