Question 1

Find the domain of $\text { f/g }$ . $f ( x ) = \sqrt { x + 7 } , \text { and } g ( x ) = \sqrt { x + 7 }$
A) $( - \infty , - 7 )$
B) $( - \infty , - 7 ]$
C) $( - \infty , 7 )$
D) $[ - 7 , \infty )$
E) $( - 7 , \infty )$

Accepted Answer

E

Question 2

Sketch the graph of the equation. $y = \sqrt { x } - 4$
A) 
B) 
C) 
D) 
E)

Accepted Answer

A

Question 3

Find the domain of $f g$. $f ( x ) = \frac { 3 x } { x - 5 } , \text { and } g ( x ) = \frac { x } { x + 1 }$
A) $( - \infty , - 1 ) \cup ( - 1,5 ) \cup ( 5 , \infty )$
B) $( - \infty , - 1 ] \cup [ 5 , \infty )$
C) $( - \infty , 1 ] \cup [ - 5 , \infty )$
D) $( - 1,5 )$
E) $( - \infty , 1 ) \cup ( - 5 , \infty )$

Accepted Answer

A

Question 4

Find a formula that expresses the fact that P(x, y) is a distance 2 away from the origin.
A) $x + y ^ { 2 } = 2$
B) $$x ^ { 2 } + y ^ { 2 } = 2$$
C) $$x + y = 2$$
D) $\sqrt { x + y } = 4$
E) $\sqrt { x ^ { 2 } + y ^ { 2 } } = 2$

Accepted Answer

The distance from the origin to the point (x, y) is given by the formula $\sqrt{x^2 + y^2}$. Setting this equal to 2 gives $\sqrt{x^2 + y^2} = 2$.

Question 5

In exercise physiology, aerobic power P is defined in terms of maximum oxygen intake. For altitudes up to 1,800 meters, aerobic power is optimal-that is, 100%. Beyond 1,800 meters, P decreases linearly from the maximum of 100% to a value near 40% at 5,000 meters. Estimate aerobic power at the altitude of 2,400 meters.
A) 72.5%
B) 93.8%
C) 72.6%
D) 88.3%
E) 88.75%

Accepted Answer

The answer of In exercise physiology, aerobic power P is...

Question 6

Find the distance d (A, B) between the points A (1, - 1) and B (8, 1).
A) 7.28
B) 6.28
C) 6.98
D) 7.78
E) 7.18

Accepted Answer

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ in a plane is given by the formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. For points A $(1, -1)$ and B $(8, 1)$, the distance is $d = \sqrt{(8 - 1)^2 + (1 - (-1))^2} = \sqrt{7^2 + 2^2} = \sqrt{49 + 4} = \sqrt{53}$, which is approximately 7.28.

Question 7

Sketch the graph of the function. $f ( x ) = 3 x ^ { 2 } - 6 x - 5$
A) 
B) 
C) 
D)

Accepted Answer

The answer of Sketch the graph of the function. \(f...

Question 8

Find the domain of $f$. $f ( x ) = \sqrt { x ^ { 2 } - 9 }$
A) $[ 3 , \infty )$
B) $( - \infty , - 3 ] \cup [ 3 , \infty )$
C) $( - \infty , - 3 ) \cup ( 3 , \infty )$
D) $( - 3,3 )$
E) $[ 9 , \infty )$

Accepted Answer

The expression under the square root, $x^2 - 9$, must be non-negative. Solving $x^2 - 9 \geq 0$ gives $x \leq -3$ or $x \geq 3$. Thus, the domain is $( - \infty , - 3 ] \cup [ 3 , \infty )$.

Question 9

If the point P (3, 1) is on the graph of a function f, find the corresponding point on the graph of the function $y = - 9 f ( x - 2 ) + 1$
A) $( 1 , - 3 )$
B) $( 5,10 )$
C) $( 5 , - 8 )$
D) $( 1 , - 8 )$
E) $( 6 , - 9 )$

Accepted Answer

The transformation $y = -9f(x-2) + 1$ shifts the graph 2 units to the right, scales it by -9, and shifts it 1 unit up. So, the point (3,1) becomes (5,-8).

Question 10

Find an equation of the line shown in the figure. 
A) $y ( x ) = - 4 + 5 x$
B) $y ( x ) = - 4 x + 5$
C) $y ( x ) = 4 x - 5$
D) $y ( x ) = 4 x + 5$
E) $y ( x ) = 5 x + 4$

Accepted Answer

The answer of Find an equation of the line shown...

Question 11

Find the distance d (A, B) between the points A (6, 6) and B (6, - 4).
A) 6
B) 0
C) 10
D) 4

Accepted Answer

Using the distance formula, we have:

d(A,B) = √[(6-6)^2 + (-4-6)^2] 
      = √[0 + 100] 
      = 10

Therefore, the distance between points A and B is 10.

Question 12

Find $( f + g ) ( 3 )$ $f ( x ) = 8 x + 2$, and $g ( x ) = x ^ { 2 }$
A) $( f + g ) ( 3 ) = 32$
B) $( f + g ) ( 3 ) = 33$
C) $( f + g ) ( 3 ) = 19$
D) $( f + g ) ( 3 ) = 39$
E) $( f + g ) ( 3 ) = 35$

Accepted Answer

The answer of Find \(( f + g ) (...

Question 13

Find the domain of $f$. $f ( x ) = \frac { 8 } { 6 x ^ { 2 } + 18 x - 108 }$
A) $( - \infty , - 6 ] \cup [ 3 , \infty )$
B) $( - \infty , - 6 ) \cup ( 3 , \infty )$
C) $( - \infty , - 6 ) \cup ( - 6,3 ) \cup ( 3 , \infty )$
D) $( - \infty , - 7 ] \cup [ 7 , \infty )$
E) $( - \infty , - 7 ) \cup ( - 7,7 ) \cup ( 7 , \infty )$

Accepted Answer

The answer of Find the domain of $f$. \(f (...

Question 14

An object is projected vertically upward with an initial velocity of $\mathcal { v } _ { 0 }$ ft/sec, and its distance $s ( t )$ in feet above the ground after $t$ seconds is given by the formula $s ( t ) = - 16 t ^ { 2 } + v _ { 0 } t$ . If the object hits the ground after $15$ seconds, find its initial velocity $\mathcal { v } _ { 0 }$ in ft/sec.
A) $209$ ft/sec
B) $226$ ft/sec
C) $17$ ft/sec
D) $240$ ft/sec
E) $256$ ft/sec

Accepted Answer

The answer of An object is projected vertically upward with...

Question 15

Find $( f \circ g ) ( x )$. $f ( x ) = x ^ { 2 } - 4 x \text {, and } g ( x ) = \sqrt { x + 9 }$
A) $( f \circ g ) ( x ) = x + 9 - 4 \sqrt { x + 9 }$
B) $( f \circ g ) ( x ) = \sqrt { x ^ { 2 } - 4 x }$
C) $( f \circ g ) ( x ) = \sqrt { x ^ { 2 } - 4 x } + 9$
D) $( f \circ g ) ( x ) = \left( x ^ { 2 } - 4 x \right) \sqrt { x + 9 }$
E) $( f \circ g ) ( x ) = \sqrt { x ^ { 2 } - 4 x + 9 }$

Accepted Answer

To find $ ( f \circ g ) ( x ) $, we substitute $ g ( x ) $ into $ f ( x ) $:$$( f \circ g ) ( x ) = f ( g ( x ) ) = f ( \sqrt { x + 9 } ) = ( \sqrt { x + 9 } ) ^ { 2 } - 4 ( \sqrt { x + 9 } ) = x + 9 - 4 \sqrt { x + 9 }$$

Question 16

Let y = f (x) be a function with domain D = [ -10, -9 ] and range R = [ -10, -7 ]. Find the domain D and range R for the function $y = | f ( x ) |$
A) $D = [ - 10 , - 9 ] , R = [ - 7,10 ]$
B) $D = [ 9,10 ] , R = [ - 10 , - 7 ]$
C) $D = [ 9,10 ] , R = [ - 10,7 ]$
D) $D = [ 9,10 ] , R = [ 7,10 ]$
E) $D = [ - 10 , - 9 ] , R = [ 7,10 ]$

Accepted Answer

The answer of Let y = f (x) be a...

Question 17

Simplify the difference quotient $\frac { f ( x + h ) - f ( x ) } { h } \text {, if } h \neq 0$ for $f ( x ) = x ^ { 2 } + 4$
A) $\frac { f ( x + h ) - f ( x ) } { h } = x + h$
B) $\frac { f ( x + h ) - f ( x ) } { h } = 2 x$
C) $\frac { f ( x + h ) - f ( x ) } { h } = 2 x + h$
D) $\frac { f ( x + h ) - f ( x ) } { h } = 2 x + h + 8$
E) $\frac { f ( x + h ) - f ( x ) } { h } = 2 x + h ^ { 2 }$

Accepted Answer

The difference quotient simplifies to $\frac{(x + h)^2 + 4 - (x^2 + 4)}{h} = \frac{x^2 + 2xh + h^2 + 4 - x^2 - 4}{h} = \frac{2xh + h^2}{h} = 2x + h$.

Question 18

Find $( f / g ) ( 5 )$. $f ( x ) = - x ^ { 2 }$, and $g ( x ) = 6 x - 2$
A) $( f / g ) ( 5 ) = - \frac { 5 } { 4 }$
B) $$( f / g ) ( 5 ) = - \frac { 25 } { 28 }$$
C) $( f / g ) ( 5 ) = \frac { 5 } { 14 }$
D) $( f / g ) ( 5 ) = \frac { 25 } { 28 }$
E) $( f / g ) ( 5 ) = - \frac { 5 } { 14 }$

Accepted Answer

The answer of Find \(( f / g ) (...

Question 19

Find an equation of the line that is tangent to the circle x² + y² = 25 at the point P ( 3, 4 ). A) $y = \frac { 3 } { 4 } x + \frac { 25 } { 4 }$ B) $y = - \frac { 4 } { 3 } x + \frac { 25 } { 3 }$ C) $y = - \frac { 3 } { 4 } x + \frac { 25 } { 4 }$ D) $y = \frac { 3 } { 4 } x - 25$ E) $y = - \frac { 3 } { 4 } x + 25$

Accepted Answer

The slope of the radius to point P(3,4) is 4/3, so the slope of the tangent is the negative reciprocal, -3/4. Using point-slope form, the equation is $y - 4 = -\frac{3}{4}(x - 3)$, which simplifies to $y = -\frac{3}{4}x + \frac{25}{4}$.

Question 20

Find two positive real numbers whose sum is 48 and whose product is a maximum.
A) 24 and 24
B) 44 and 4
C) 2 and 46
D) 27 and 21
E) 31 and 17

Accepted Answer

The product of two numbers with a fixed sum is maximized when the numbers are equal, so 24 and 24 (which sum to 48) will yield the maximum product.

Quiz 2: Functions and Graphs

Find the domain of $\text { f/g }$ . $f ( x ) = \sqrt { x + 7 } , \text { and } g ( x ) = \sqrt { x + 7 }$

Sketch the graph of the equation. $y = \sqrt { x } - 4$

Find the domain of $f g$ . $f ( x ) = \frac { 3 x } { x - 5 } , \text { and } g ( x ) = \frac { x } { x + 1 }$

Find a formula that expresses the fact that P(x, y) is a distance 2 away from the origin.

Find the distance d (A, B) between the points A (1, - 1) and B (8, 1) .

Sketch the graph of the function. $f ( x ) = 3 x ^ { 2 } - 6 x - 5$

Find the domain of $f$ . $f ( x ) = \sqrt { x ^ { 2 } - 9 }$

If the point P (3, 1) is on the graph of a function f, find the corresponding point on the graph of the function $y = - 9 f ( x - 2 ) + 1$

Find an equation of the line shown in the figure.

Find the distance d (A, B) between the points A (6, 6) and B (6, - 4) .

Find $( f + g ) ( 3 )$ $f ( x ) = 8 x + 2$ , and $g ( x ) = x ^ { 2 }$

Find the domain of $f$ . $f ( x ) = \frac { 8 } { 6 x ^ { 2 } + 18 x - 108 }$

Find $( f \circ g ) ( x )$ . $f ( x ) = x ^ { 2 } - 4 x \text {, and } g ( x ) = \sqrt { x + 9 }$

Let y = f (x) be a function with domain D = [ -10, -9 ] and range R = [ -10, -7 ]. Find the domain D and range R for the function $y = | f ( x ) |$

Simplify the difference quotient $\frac { f ( x + h ) - f ( x ) } { h } \text {, if } h \neq 0$ for $f ( x ) = x ^ { 2 } + 4$

Find $( f / g ) ( 5 )$ . $f ( x ) = - x ^ { 2 }$ , and $g ( x ) = 6 x - 2$

Find an equation of the line that is tangent to the circle x² + y² = 25 at the point P ( 3, 4 ) .

Find two positive real numbers whose sum is 48 and whose product is a maximum.

Quiz 2: Functions and Graphs

Find the domain of f/g \text { f/g } f/g . f(x) =x+7, and g(x) =x+7f ( x ) = \sqrt { x + 7 } , \text { and } g ( x ) = \sqrt { x + 7 }f(x) =x+7​, and g(x) =x+7​

Sketch the graph of the equation. y=x−4y = \sqrt { x } - 4y=x​−4

Find the domain of fgf gfg . f(x) =3xx−5, and g(x) =xx+1f ( x ) = \frac { 3 x } { x - 5 } , \text { and } g ( x ) = \frac { x } { x + 1 }f(x) =x−53x​, and g(x) =x+1x​

Find a formula that expresses the fact that P(x, y) is a distance 2 away from the origin.

Find the distance d (A, B) between the points A (1, - 1) and B (8, 1) .

Sketch the graph of the function. f(x) =3x2−6x−5f ( x ) = 3 x ^ { 2 } - 6 x - 5f(x) =3x2−6x−5

Find the domain of fff . f(x) =x2−9f ( x ) = \sqrt { x ^ { 2 } - 9 }f(x) =x2−9​

If the point P (3, 1) is on the graph of a function f, find the corresponding point on the graph of the function y=−9f(x−2) +1y = - 9 f ( x - 2 ) + 1y=−9f(x−2) +1

Find an equation of the line shown in the figure.

Find the distance d (A, B) between the points A (6, 6) and B (6, - 4) .

Find (f+g) (3) ( f + g ) ( 3 ) (f+g) (3) f(x) =8x+2f ( x ) = 8 x + 2f(x) =8x+2 , and g(x) =x2g ( x ) = x ^ { 2 }g(x) =x2

Find the domain of fff . f(x) =86x2+18x−108f ( x ) = \frac { 8 } { 6 x ^ { 2 } + 18 x - 108 }f(x) =6x2+18x−1088​

Find (f∘g) (x) ( f \circ g ) ( x ) (f∘g) (x) . f(x) =x2−4x, and g(x) =x+9f ( x ) = x ^ { 2 } - 4 x \text {, and } g ( x ) = \sqrt { x + 9 }f(x) =x2−4x, and g(x) =x+9​

Let y = f (x) be a function with domain D = [ -10, -9 ] and range R = [ -10, -7 ]. Find the domain D and range R for the function y=∣f(x) ∣y = | f ( x ) |y=∣f(x) ∣

Simplify the difference quotient f(x+h) −f(x) h, if h≠0\frac { f ( x + h ) - f ( x ) } { h } \text {, if } h \neq 0hf(x+h) −f(x) ​, if h=0 for f(x) =x2+4f ( x ) = x ^ { 2 } + 4f(x) =x2+4

Find (f/g) (5) ( f / g ) ( 5 ) (f/g) (5) . f(x) =−x2f ( x ) = - x ^ { 2 }f(x) =−x2 , and g(x) =6x−2g ( x ) = 6 x - 2g(x) =6x−2

Find an equation of the line that is tangent to the circle x 2 + y 2 = 25 at the point P ( 3, 4 ) .