Exam 4: Additional Topics With Functions

Find the specified domain and express it in interval notation. -For $f ( x ) = x ^ { 2 } - 81$ and $g ( x ) = 2 x + 3$ , what is the domain of $\left( \frac { g } { f } \right) ( x )$ ?

Explain why the functi $f ( x ) = ( x - 1 ) ^ { 2 } + 7$ is not a one-to-one function.

The price per unit of a product is $\mathrm { p }$ and the number of units of the product is denoted by q. The demand function for this commodity is given by $p = \frac { 4000 } { q + 60 }$ . Describe the transformations needed to obtain the graph of this function from the graph of $p = \frac { 1 } { q }$ .

Graph the given function as a solid line (or curve) and its inverse as a dashed line (or curve) on the same set of axes. - $f ( x ) = - \sqrt { x + 6 }$

Determine the values of x for which f(x) and g(x) are inverse functions. - $f ( x ) = ( x - 8 ) ^ { 2 } , g ( x ) = \sqrt { x } + 8$

Fill in each blank with the appropriate response. -The graph of $\mathrm { y } = 2 \sqrt [ 3 ] { \mathrm { x } - 1 } - 3$ can be obtained from the graph of $\mathrm { y } = \sqrt [ 3 ] { \mathrm { x } }$ by shifting horizontallyــــــــــ units to theـــــــــ , vertically stretching by a factor ofـــــــــــــــ and shifting vertically ــــــــــــunits in the directionـــــــــــــــــ.

Find the inverse of the function. - $f ( x ) = \frac { - 3 x + 3 } { 8 x - 2 }$

Solve the inequality by using algebraic and graphical methods. Round to three decimal places, if necessary. - $| x - 7 | > 9$

A stone is thrown into a pond. A circular ripple is spreading over the pond in such a way that the radius is increasing at the rate of 2.6 feet per second. Find a function, r(t) , for the radius in terms of t. Find a function, A(r) , for the area of the ripple in terms of r . Find $( \mathrm { A } \cdot \mathrm { r } ) ( \mathrm { t } )$

Solve the inequality. - $x ^ { 2 } + 7 x \leq - 3 x - 21$

Solve the inequality. - $4 x ^ { 2 } - 32 x + 60 < 0$

Acme Communication finds that the total revenue function associated with producing a new type of cellular phone is $R ( x ) = 213 x - x ^ { 2 }$ . and the total cost function is C(x)=8000+6 x , where x represents the number of units of cellular phones produced. Find the total profit function, P(x) .

Fill in each blank with the appropriate response. -The graph of $y = - 5 ( x + 3 ) ^ { 2 } - 7$ can be obtained from the graph of $y = x ^ { 2 }$ by shifting horizontallyــــــــ units to the , vertically stretching by a factor ofــــــــ , reflecting across theـــــــــــ -axis, and shifting vertically ــــــــــــunits in the ــــــــــــdirection.

The graphs of two functions f(x) and g(x) are given. -Solve $f ( x ) \geq g ( x )$ .

Determine whether the function is even, odd, or neither. - $f ( x ) = 5 x ^ { 2 } + 4$

Provide an appropriate response. -Your classmate missed the lecture on how to find the inverse of a one-to-one function. When your classmate calls you on the phone for an explanation, what will you say to explain the process?

Decide whether or not the functions are inverses of each other. - $f ( x ) = \frac { 3 } { x + 2 } , g ( x ) = \frac { 2 x + 3 } { x }$

Determine whether (f(g(x)) $( f ( g ( x ) ) = x \text { and whether } ( g ( f ( x ) ) = x$ - $f ( x ) = \frac { 1 } { x } , g ( x ) = x$

The year y when sales were s million dollars for a particular electronics company can be modeled by the radical equation $y = 1.2 \sqrt { s - 4 } - 6$ , where $y = 1$ represents 2010 , and so on. Describe the transformations needed to obtain the graph of this function from the graph of $y = \sqrt { s }$ .