Question 1

The graph of the function $g ( x )$ is a translation of the graph of $f ( x ) = x ^ { 3 }$. Graph the function. $g ( x ) = ( x - 2 ) ^ { 3 } - 5$
A)

B)

C)

D)

Accepted Answer

To translate a function, we need to shift the graph horizontally and/or vertically. In this case, we are shifting the graph to the right by 2 units and down by 5 units. 
Therefore, the graph of $g(x) = (x-2)^3 - 5$ will be the graph of $f(x) = x^3$ shifted 2 units to the right and 5 units down. 
Choice B matches this description.

Question 2

Graph the function. $f ( x ) = \frac { x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 4 x + 4 }$
A)

B)

C)

D)

Accepted Answer

The answer of Graph the function. \(f ( x )...

Question 3

The graph of the function $g ( x )$ is a translation of the graph of $f ( x ) = x ^ { 3 }$. Graph the function.
A)

B)

C)

$\mathrm { d } .$

Accepted Answer

The answer of The graph of the function \(g (...

Question 4

Find the inverse of the one-to-one function. $$y = 3 x + 4$$
A) $y = \frac { x + 4 } { 3 }$
B) $y = \frac { x - 3 } { 4 }$
C) none of the above
D) $y = \frac { x - 4 } { 3 }$
E) $y = \frac { 3 } { x - 4 }$

Accepted Answer

To find the inverse, we can first switch $x$ and $y$ and solve for $y$: $$x = 3y + 4 \Rightarrow y = \frac{x-4}{3}$$ Therefore, the inverse function is $y = \frac{x-4}{3}$, which is choice $\boxed{\text{(D)}}$.

Question 5

Let $f ( x ) = 2 x - 1 , g ( x ) = 3 x - 2$. Find the domain of the function.
$$( f - g ) ( x )$$
A) $\quad ( - \infty , 0 )$
B) $\quad ( - \infty , 0 ]$
C) $( - \infty , \infty )$
D) $\quad ( 0 , \infty )$
E) $\quad [ 0 , \infty )$

Accepted Answer

The domain of $f(x) = 2x - 1$ and $g(x) = 3x - 2$ is all real numbers, $(-\infty, \infty)$, because there are no restrictions (like division by zero or square roots of negative numbers) in either function. When subtracting these two functions, $(f - g)(x)$, the domain remains all real numbers.

Question 6

Find the inverse of the one-to-one function. $$y = 8 x$$
A) $\quad y = \frac { 8 } { x }$
B) $\quad y = \frac { x } { 8 }$
C) $\quad y = 64 x$
D) $\quad y = 8 x ^ { 2 }$
E) $y = 8 x$

Accepted Answer

To find the inverse of a function, we switch the roles of $x$ and $y$ and solve for $y$. Starting with the function $y=8x$, we switch $x$ and $y$ to get $x=8y$. Solving for $y$, we get $y=\frac{x}{8}$. Therefore, the inverse of the function is $y=\frac{x}{8}$.

Question 7

A pizzeria charges $8.50 plus $0.90 per topping for a large pizza.Find the cost of a pizza that has 5 toppings. 
A) $ \$ 13.00$
B) $\$ 3.75$
C) $\$ 14.35$
D) $\$ 4.00$
E) $\$ 14.00$

Accepted Answer

The answer of A pizzeria charges $8.50 plus $0.90 per...

Question 8

The function $f ( x ) = \sqrt { x ^ { 2 } - 9 }$ is one-to-one on the domain $x \leq - \sqrt { 9 }$. Find $f ^ { - 1 } ( x )$.
A) $\quad f ^ { - 1 } ( x ) = x ^ { 2 } + 9$
B) $f ^ { - 1 } ( x ) = - \sqrt { x ^ { 2 } + 9 }$
C) $f ^ { - 1 } ( x ) = \frac { 1 } { \sqrt { x ^ { 2 } - 9 } }$
D) $\quad f ^ { - 1 } ( x ) = \sqrt { x ^ { 2 } + 9 }$
E) $\quad f ^ { - 1 } ( x ) = - \sqrt { x ^ { 2 } - 9 }$

Accepted Answer

The inverse function is found by solving $y = \sqrt{x^2 - 9}$ for $x$, which gives $x = -\sqrt{y^2 + 9}$ since the domain is $x \leq -3$.

Question 9

Graph the rational function. $$f ( x ) = \frac { 2 x ^ { 3 } + 4 x } { 5 x }$$
Note that the numerator and denominator of the fraction share a common factor.
A)

B)

C)

$\mathrm { d } .$

Accepted Answer

The rational function can be simplified by factoring out a common factor of $2x$ in the numerator and cancelling with the $5x$ in the denominator, leaving us with $f(x) = \frac{2x^2}{5}$. This is a parabola with vertex at the origin, opening upwards, with a vertical stretch by a factor of $\frac{2}{5}$. The graph passes through the origin and has $x$-intercepts at $x=0$. The $y$-intercept occurs when $x=0$, which is $y=f(0)=0$. The graph is shown in choice B.

Question 10

Find the range of the function $f ( x ) = \frac { 4 x } { x - 2 }$ by finding the domain of $f ^ { - 1 } ( x )$.
A) $\quad ( - \infty , 2 ) \cup ( 2 , \infty )$
B) $\quad ( - \infty , 4 ) \cup ( 4 , \infty )$
C) $\quad ( - \infty , \infty )$
D) $\quad ( - \infty , 0 ) \cup ( 0 , \infty )$
E) $\quad ( 0 , \infty )$

Accepted Answer

To find the range of $f(x) = \frac{4x}{x-2}$, we find the domain of its inverse $f^{-1}(x)$. The function $f(x)$ has a vertical asymptote at $x = 2$ and no horizontal asymptote, as the degree of the numerator and denominator are the same but the function approaches infinity as $x$ approaches 2 from either side. The function can take all real values except for the value it approaches as $x$ goes to infinity, which is the coefficient ratio $4/1 = 4$. Thus, the range of $f(x)$, and hence the domain of $f^{-1}(x)$, is all real numbers except 4, or $(-\infty, 4) \cup (4, \infty)$.

Question 11

Determine whether the function is one-to-one. $$y = \frac { 2 } { 9 } x$$
A) No, it isn't one-to-one.
B) Yes, it is one-to-one.

Accepted Answer

To determine if a function is one-to-one, we need to check if different inputs (x-values) produce different outputs (y-values). In this case, since every value of x corresponds to one and only one value of y, the function is one-to-one. Therefore, the correct answer is B) Yes, it is one-to-one.

Question 12

The graph of the function $f ( x )$ is a translation of the graph of $y = ( x - 1 ) ^ { 10 }$. Graph the function. $f ( x ) = ( - x - 1 ) ^ { 10 }$
A)

B)

C)

D)

Accepted Answer

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

Question 13

Let $f ( x ) = \sqrt { x + 1 } , g ( x ) = x ^ { 2 } - 1$. Find the domain of the composite function. $( g \circ f ) ( x )$
A) $[ - 1 , \infty )$
B) $\quad ( - \infty , - 1 ]$
C) $\quad ( - \infty , - 1 )$
D) $( - \infty , \infty )$
E) $( - 1 , \infty )$

Accepted Answer

The answer of Let \(f ( x ) = \sqrt...

Question 14

Let $f ( x ) = \sqrt { x } , g ( x ) = x + 2$. Find the composite function which expresses the given correspondence correctly.
$\sqrt [ 4 ] { x }$
A) $\quad ( f \circ f ) ( x )$
B) $\quad ( f \circ g ) ( x )$
C) $( g \circ f ) ( x )$
D) none of the above
E) $( g \circ g ) ( x )$

Accepted Answer

The correct composite function is $( f \circ f ) ( x )$, because applying $f(x) = \sqrt{x}$ twice results in $\sqrt{\sqrt{x}} = \sqrt[4]{x}$.

Question 15

Let $f ( x ) = 3 x , g ( x ) = x + 1$. Find the composite function.
$$( g \circ g ) ( x )$$
A) $\quad ( g \circ g ) ( x ) = 9 x$
B) none of the above
C) $( g \circ g ) ( x ) = 3 x + 1$
D) $( g \circ g ) ( x ) = x + 2$
E) $\quad ( g \circ g ) ( x ) = 3 x + 3$

Accepted Answer

$(g \circ g)(x) = g(g(x)) = g(x+1) = (x+1) + 1 = x+2$

Quiz 4: Functions Functions and Function Notation

The graph of the function $g ( x )$ is a translation of the graph of $f ( x ) = x ^ { 3 }$ . Graph the function. $g ( x ) = ( x - 2 ) ^ { 3 } - 5$

Graph the function. $f ( x ) = \frac { x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 4 x + 4 }$

The graph of the function $g ( x )$ is a translation of the graph of $f ( x ) = x ^ { 3 }$ . Graph the function.

Find the inverse of the one-to-one function. $y = 3 x + 4$

Let $f ( x ) = 2 x - 1 , g ( x ) = 3 x - 2$ . Find the domain of the function. $( f - g ) ( x )$

Find the inverse of the one-to-one function. $y = 8 x$

A pizzeria charges $8.50 plus $0.90 per topping for a large pizza.Find the cost of a pizza that has 5 toppings.

The function $f ( x ) = \sqrt { x ^ { 2 } - 9 }$ is one-to-one on the domain $x \leq - \sqrt { 9 }$ . Find $f ^ { - 1 } ( x )$ .

Graph the rational function. $f ( x ) = \frac { 2 x ^ { 3 } + 4 x } { 5 x }$ Note that the numerator and denominator of the fraction share a common factor.

Find the range of the function $f ( x ) = \frac { 4 x } { x - 2 }$ by finding the domain of $f ^ { - 1 } ( x )$ .

Determine whether the function is one-to-one. $y = \frac { 2 } { 9 } x$

The graph of the function $f ( x )$ is a translation of the graph of $y = ( x - 1 ) ^ { 10 }$ . Graph the function. $f ( x ) = ( - x - 1 ) ^ { 10 }$

Let $f ( x ) = \sqrt { x + 1 } , g ( x ) = x ^ { 2 } - 1$ . Find the domain of the composite function. $( g \circ f ) ( x )$

Let $f ( x ) = \sqrt { x } , g ( x ) = x + 2$ . Find the composite function which expresses the given correspondence correctly. $\sqrt [ 4 ] { x }$

Let $f ( x ) = 3 x , g ( x ) = x + 1$ . Find the composite function. $( g \circ g ) ( x )$

Quiz 4: Functions Functions and Function Notation

The graph of the function g(x) g ( x ) g(x) is a translation of the graph of f(x) =x3f ( x ) = x ^ { 3 }f(x) =x3 . Graph the function. g(x) =(x−2) 3−5g ( x ) = ( x - 2 ) ^ { 3 } - 5g(x) =(x−2) 3−5

Graph the function. f(x) =x2−3x−2x2−4x+4f ( x ) = \frac { x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 4 x + 4 }f(x) =x2−4x+4x2−3x−2​

The graph of the function g(x) g ( x ) g(x) is a translation of the graph of f(x) =x3f ( x ) = x ^ { 3 }f(x) =x3 . Graph the function.

Find the inverse of the one-to-one function. y=3x+4y = 3 x + 4y=3x+4

Let f(x) =2x−1,g(x) =3x−2f ( x ) = 2 x - 1 , g ( x ) = 3 x - 2f(x) =2x−1,g(x) =3x−2 . Find the domain of the function. (f−g) (x) ( f - g ) ( x ) (f−g) (x)

Find the inverse of the one-to-one function. y=8xy = 8 xy=8x

A pizzeria charges $8.50 plus $0.90 per topping for a large pizza.Find the cost of a pizza that has 5 toppings.

The function f(x) =x2−9f ( x ) = \sqrt { x ^ { 2 } - 9 }f(x) =x2−9​ is one-to-one on the domain x≤−9x \leq - \sqrt { 9 }x≤−9​ . Find f−1(x) f ^ { - 1 } ( x ) f−1(x) .

Graph the rational function. f(x) =2x3+4x5xf ( x ) = \frac { 2 x ^ { 3 } + 4 x } { 5 x }f(x) =5x2x3+4x​ Note that the numerator and denominator of the fraction share a common factor.

Find the range of the function f(x) =4xx−2f ( x ) = \frac { 4 x } { x - 2 }f(x) =x−24x​ by finding the domain of f−1(x) f ^ { - 1 } ( x ) f−1(x) .

Determine whether the function is one-to-one. y=29xy = \frac { 2 } { 9 } xy=92​x

The graph of the function f(x) f ( x ) f(x) is a translation of the graph of y=(x−1) 10y = ( x - 1 ) ^ { 10 }y=(x−1) 10 . Graph the function. f(x) =(−x−1) 10f ( x ) = ( - x - 1 ) ^ { 10 }f(x) =(−x−1) 10

Let f(x) =x+1,g(x) =x2−1f ( x ) = \sqrt { x + 1 } , g ( x ) = x ^ { 2 } - 1f(x) =x+1​,g(x) =x2−1 . Find the domain of the composite function. (g∘f) (x) ( g \circ f ) ( x ) (g∘f) (x)

Let f(x) =x,g(x) =x+2f ( x ) = \sqrt { x } , g ( x ) = x + 2f(x) =x​,g(x) =x+2 . Find the composite function which expresses the given correspondence correctly. x4\sqrt [ 4 ] { x }4x​

Let f(x) =3x,g(x) =x+1f ( x ) = 3 x , g ( x ) = x + 1f(x) =3x,g(x) =x+1 . Find the composite function. (g∘g) (x) ( g \circ g ) ( x ) (g∘g) (x)