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

ƒ(x) = (x + 1)³

Solving equations Solve the following equations.

log₈ x = 1/3

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

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

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

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

cos⁻¹ √3/2

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 )

Intersection problems Find the following points of intersection.

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

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

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)

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.

${\log }_{6}60$

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|

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 .

Let ƒ(x) = 1/ (x³+1).

Compute ƒ(2) and ƒ(y²).

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).

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

Evaluate each expression without a calculator.

a. log₁₀ 1000

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

$f\left(x\right)=\frac{2}{x^2+1}$, for $x\geq{0}$

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))

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

logbx²

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

(g o ƒ ) (x) = x⁴ + 3

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

Convert the following expressions to the indicated base.

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

Solve each equation.

$7^{y-3}=50$

Finding inverses Find the inverse function.

ƒ(x) = 3x² + 1, for x ≤ 0

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

ƒ(x) = x⁴

Solving equations Solve the following equations.

ln x= -1

Graph the following functions.

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State whether the functions represented by graphs A , B , C and ​​ in the figure are even, odd, or neither. <IMAGE>

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))