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

ƒ(x) = (x)/(x+1)

Composite functions and notation

Let ƒ(x)= x² - 4 , g(x) = x³ and F(x) = 1/(x-3).

Simplify or evaluate the following expressions.

g(ƒ(u))

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

ƒ(x) = (x + 1)³

Defining piecewise functions Write a definition of the function whose graph is given <IMAGE>

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

ƒ(x) = 2 / ( x² + 2)

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.

$\text{ƒ}\left(x\right)=x^4+5x^2-12$

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)

Solving equations Solve the following equations.

log₈ x = 1/3

Intersection problems Find the following points of intersection.

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

Finding inverses Find the inverse function.

ƒ(x) = 3x - 4

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

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

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

cos (cos⁻¹ ( -1 ))

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

Solving equations Solve the following equations.

3(ˣ³⁻⁴) = 15

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 the following equations.

log₁₀ x= 3

Solving equations Solve the following equations.

5(ˣ³) = 29

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

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

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.

Let ƒ(x) = 1/ (x³+1).

Compute ƒ(2) and ƒ(y²).

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

ƒ(x) = 2x² -3x +1

Solve each equation.

$7^{y-3}=50$

Convert the following expressions to the indicated base.

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

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

ƒ(x) = x² -2x + 6

Finding inverses Find the inverse function.

ƒ(x) = 3x² + 1, for x ≤ 0

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

logbx²

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) = (x³ - 5)¹⁰

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 $x\gt{2}$

Write the following logarithms in terms of the natural logarithm. Then use a calculator to find the value of the logarithm, rounding your result to four decimal places.

${\log }_{6}60$

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( g⁻¹(4))