Exam 2: Functions

Solve the problem. - $\text { Let } f ( x ) = x - 6 \text { and } g ( x ) = x ^ { 2 } \text {. Graph } f \text { and } g \text { together with } f \text { og and } g \circ \text { f. }$ $Solve the problem. - \text { Let } f ( x ) = x - 6 \text { and } g ( x ) = x ^ { 2 } \text {. Graph } f \text { and } g \text { together with } f \text { og and } g \circ \text { f. }$

(Essay)

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Question 181

Find the domain and range for the indicated function. - $f ( x ) = \sqrt { x + 5 , } \quad g ( x ) = \sqrt { x - 5 }$ $\mathrm { f } - \mathrm { g }$

(Multiple Choice)

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Question 182

Solve the problem. -Let $f ( x ) = \sqrt { x - 2 }$ . Find a function $y = g ( x )$ so that $( f \circ g ) ( x ) = \sqrt { x ^ { 2 } - 2 }$ .

(Multiple Choice)

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Question 183

Solve the problem. -The accompanying figure shows the graph of $y = - x ^ { 2 }$ shifted to a new position. Write the equation for the new graph. $Solve the problem. -The accompanying figure shows the graph of y = - x ^ { 2 } shifted to a new position. Write the equation for the new graph.$

(Multiple Choice)

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Question 184

Simplify the expression. - $\ln \left( e ^ { 8 x } \right)$

(Multiple Choice)

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Question 185

Solve the problem. -If $f ( x ) = - 3 x + 2$ and $g ( x ) = 2 x + 9$ , find $g ( f ( x ) )$ .

(Multiple Choice)

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Question 186

Express the given quantity in terms of sin x or cos x. - $\sin \left( \frac { 7 \pi } { 2 } - x \right)$

(Multiple Choice)

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Question 187

Find the domain and range of the inverse of the given function. - $f ( x ) = \frac { 1 } { 4 } x - 6$

(Multiple Choice)

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Question 188

Find the exact function value. - $\arccos ( 0 )$

(Multiple Choice)

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Question 189

Determine from its graph if the function is one-to-one. - $f ( x ) = \left\{ \begin{array} { l l } - x - 3 , & x < 1 \\5 , & x \geq 1\end{array} \right.$

(True/False)

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Question 190

Find a formula for the function graphed. -

(Multiple Choice)

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Question 191

Express the following logarithm as specified. - $\ln 4.5$ in terms of $\ln 2$ and $\ln 3$

(Multiple Choice)

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Question 192

Provide an appropriate response. Consider the function y Can x be greater than 1? - $y = \sqrt { 1 - \frac { 1 } { x } }$

(True/False)

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Question 193

Graph the function. - $\text { Graph the upper half of the circle defined by the equation } x ^ { 2 } + y ^ { 2 } - 10 x - 8 y + 25 = 0 \text {. }$ $Graph the function. - \text { Graph the upper half of the circle defined by the equation } x ^ { 2 } + y ^ { 2 } - 10 x - 8 y + 25 = 0 \text {. }$

(Multiple Choice)

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Question 194

Use a graphing calculator or computer to determine which of the given viewing windows displays the most appropriate graph of the specified function. - $f ( x ) = x ^ { 2 } + \frac { 1 } { 10 } \cos 70 x$

(Multiple Choice)

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Question 195

Solve the problem. -The kinetic energy K of a mass is proportional to the square of its velocity v. If K = 4320 joules when v = 12 m/sec, what is K when v = 8 m/sec?

(Multiple Choice)

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Question 196

Assume that f is an even function, g is an odd function, and both f and g are defined on the entire real line. State whether the combination of functions (where defined) is even or odd. - $g \circ g$

(Multiple Choice)

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Question 197

Solve the problem. -Let $f ( x ) = \frac { x } { x - 4 }$ . Find a function $y = g ( x )$ so that $( f \circ g ) ( x ) = x$ .

(Multiple Choice)

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Question 198

Determine if the function is even, odd, or neither. - $g ( x ) = \frac { - 6 x } { x ^ { 2 } - 3 }$

(Multiple Choice)

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Question 199

Find a formula for the function graphed. -

(Multiple Choice)

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Question 200

Find the domain and range for the indicated function. - $f ( x ) = \sqrt { x + 5 , } \quad g ( x ) = \sqrt { x - 5 }$ $\mathrm { f } - \mathrm { g }$

Solve the problem. -Let $f ( x ) = \sqrt { x - 2 }$ . Find a function $y = g ( x )$ so that $( f \circ g ) ( x ) = \sqrt { x ^ { 2 } - 2 }$ .

Solve the problem. -The accompanying figure shows the graph of $y = - x ^ { 2 }$ shifted to a new position. Write the equation for the new graph. $Solve the problem. -The accompanying figure shows the graph of y = - x ^ { 2 } shifted to a new position. Write the equation for the new graph.$

Simplify the expression. - $\ln \left( e ^ { 8 x } \right)$

Solve the problem. -If $f ( x ) = - 3 x + 2$ and $g ( x ) = 2 x + 9$ , find $g ( f ( x ) )$ .

Express the given quantity in terms of sin x or cos x. - $\sin \left( \frac { 7 \pi } { 2 } - x \right)$

Find the domain and range of the inverse of the given function. - $f ( x ) = \frac { 1 } { 4 } x - 6$

Find the exact function value. - $\arccos ( 0 )$

Determine from its graph if the function is one-to-one. - $f ( x ) = \left\{ \begin{array} { l l } - x - 3 , & x < 1 \\5 , & x \geq 1\end{array} \right.$

Find a formula for the function graphed. -

Express the following logarithm as specified. - $\ln 4.5$ in terms of $\ln 2$ and $\ln 3$

Provide an appropriate response. Consider the function y Can x be greater than 1? - $y = \sqrt { 1 - \frac { 1 } { x } }$

Graph the function. - $\text { Graph the upper half of the circle defined by the equation } x ^ { 2 } + y ^ { 2 } - 10 x - 8 y + 25 = 0 \text {. }$ $Graph the function. - \text { Graph the upper half of the circle defined by the equation } x ^ { 2 } + y ^ { 2 } - 10 x - 8 y + 25 = 0 \text {. }$

Use a graphing calculator or computer to determine which of the given viewing windows displays the most appropriate graph of the specified function. - $f ( x ) = x ^ { 2 } + \frac { 1 } { 10 } \cos 70 x$

Solve the problem. -The kinetic energy K of a mass is proportional to the square of its velocity v. If K = 4320 joules when v = 12 m/sec, what is K when v = 8 m/sec?

Assume that f is an even function, g is an odd function, and both f and g are defined on the entire real line. State whether the combination of functions (where defined) is even or odd. - $g \circ g$

Solve the problem. -Let $f ( x ) = \frac { x } { x - 4 }$ . Find a function $y = g ( x )$ so that $( f \circ g ) ( x ) = x$ .

Determine if the function is even, odd, or neither. - $g ( x ) = \frac { - 6 x } { x ^ { 2 } - 3 }$

Find a formula for the function graphed. -

Limits and Continuity

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Integrals

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Integrals and Transcendental Functions

Techniques of Integration

First-Order Differential Equations

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Parametric Equations and Polar Coordinates

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Exam 2: Functions

Find the domain and range for the indicated function. - f(x)=x+5,g(x)=x−5f ( x ) = \sqrt { x + 5 , } \quad g ( x ) = \sqrt { x - 5 }f(x)=x+5,​g(x)=x−5​ f−g\mathrm { f } - \mathrm { g }f−g

Solve the problem. -Let f(x)=x−2f ( x ) = \sqrt { x - 2 }f(x)=x−2​ . Find a function y=g(x)y = g ( x )y=g(x) so that (f∘g)(x)=x2−2( f \circ g ) ( x ) = \sqrt { x ^ { 2 } - 2 }(f∘g)(x)=x2−2​ .

Solve the problem. -The accompanying figure shows the graph of y=−x2y = - x ^ { 2 }y=−x2 shifted to a new position. Write the equation for the new graph.

Simplify the expression. - ln⁡(e8x)\ln \left( e ^ { 8 x } \right)ln(e8x)

Solve the problem. -If f(x)=−3x+2f ( x ) = - 3 x + 2f(x)=−3x+2 and g(x)=2x+9g ( x ) = 2 x + 9g(x)=2x+9 , find g(f(x))g ( f ( x ) )g(f(x)) .

Express the given quantity in terms of sin x or cos x. - sin⁡(7π2−x)\sin \left( \frac { 7 \pi } { 2 } - x \right)sin(27π​−x)

Find the domain and range of the inverse of the given function. - f(x)=14x−6f ( x ) = \frac { 1 } { 4 } x - 6f(x)=41​x−6

Find the exact function value. - arccos⁡(0)\arccos ( 0 )arccos(0)

Determine from its graph if the function is one-to-one. - f(x)={−x−3,x<15,x≥1f ( x ) = \left\{ \begin{array} { l l } - x - 3 , & x < 1 \\5 , & x \geq 1\end{array} \right.f(x)={−x−3,5,​x<1x≥1​

Find a formula for the function graphed. -

Express the following logarithm as specified. - ln⁡4.5\ln 4.5ln4.5 in terms of ln⁡2\ln 2ln2 and ln⁡3\ln 3ln3

Provide an appropriate response. Consider the function y Can x be greater than 1? - y=1−1xy = \sqrt { 1 - \frac { 1 } { x } }y=1−x1​​

Use a graphing calculator or computer to determine which of the given viewing windows displays the most appropriate graph of the specified function. - f(x)=x2+110cos⁡70xf ( x ) = x ^ { 2 } + \frac { 1 } { 10 } \cos 70 xf(x)=x2+101​cos70x

Solve the problem. -The kinetic energy K of a mass is proportional to the square of its velocity v. If K = 4320 joules when v = 12 m/sec, what is K when v = 8 m/sec?

Assume that f is an even function, g is an odd function, and both f and g are defined on the entire real line. State whether the combination of functions (where defined) is even or odd. - g∘gg \circ gg∘g

Solve the problem. -Let f(x)=xx−4f ( x ) = \frac { x } { x - 4 }f(x)=x−4x​ . Find a function y=g(x)y = g ( x )y=g(x) so that (f∘g)(x)=x( f \circ g ) ( x ) = x(f∘g)(x)=x .

Determine if the function is even, odd, or neither. - g(x)=−6xx2−3g ( x ) = \frac { - 6 x } { x ^ { 2 } - 3 }g(x)=x2−3−6x​

Find a formula for the function graphed. -

Limits and Continuity

Derivatives

Applications of Derivatives

Integrals

Applications of Definite Integrals

Integrals and Transcendental Functions

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions and Motion in Space

Partial Derivatives

Multiple Integrals

Integrals and Vector Fields

Filters

Find the domain and range for the indicated function. - $f ( x ) = \sqrt { x + 5 , } \quad g ( x ) = \sqrt { x - 5 }$ $\mathrm { f } - \mathrm { g }$

Solve the problem. -Let $f ( x ) = \sqrt { x - 2 }$ . Find a function $y = g ( x )$ so that $( f \circ g ) ( x ) = \sqrt { x ^ { 2 } - 2 }$ .

Solve the problem. -The accompanying figure shows the graph of $y = - x ^ { 2 }$ shifted to a new position. Write the equation for the new graph. $Solve the problem. -The accompanying figure shows the graph of y = - x ^ { 2 } shifted to a new position. Write the equation for the new graph.$

Simplify the expression. - $\ln \left( e ^ { 8 x } \right)$

Solve the problem. -If $f ( x ) = - 3 x + 2$ and $g ( x ) = 2 x + 9$ , find $g ( f ( x ) )$ .

Express the given quantity in terms of sin x or cos x. - $\sin \left( \frac { 7 \pi } { 2 } - x \right)$

Find the domain and range of the inverse of the given function. - $f ( x ) = \frac { 1 } { 4 } x - 6$

Find the exact function value. - $\arccos ( 0 )$

Determine from its graph if the function is one-to-one. - $f ( x ) = \left\{ \begin{array} { l l } - x - 3 , & x < 1 \\5 , & x \geq 1\end{array} \right.$

Express the following logarithm as specified. - $\ln 4.5$ in terms of $\ln 2$ and $\ln 3$

Provide an appropriate response. Consider the function y Can x be greater than 1? - $y = \sqrt { 1 - \frac { 1 } { x } }$

Use a graphing calculator or computer to determine which of the given viewing windows displays the most appropriate graph of the specified function. - $f ( x ) = x ^ { 2 } + \frac { 1 } { 10 } \cos 70 x$

Assume that f is an even function, g is an odd function, and both f and g are defined on the entire real line. State whether the combination of functions (where defined) is even or odd. - $g \circ g$

Solve the problem. -Let $f ( x ) = \frac { x } { x - 4 }$ . Find a function $y = g ( x )$ so that $( f \circ g ) ( x ) = x$ .

Determine if the function is even, odd, or neither. - $g ( x ) = \frac { - 6 x } { x ^ { 2 } - 3 }$