Question 1

Let $f ( x ) = 3 x - 4$ and $g ( x ) = x ^ { 2 } - 1$ . Find: $( f \circ g ) ( x )$
A) $3 x ^ { 2 } - 5$
B) $3 x ^ { 2 } - 7$
C) $9 x ^ { 2 } - 24 x + 15$
D) $3 x ^ { 2 } - 3$

Accepted Answer

B

Question 2

If $f ( x ) = \sqrt { x }$ and $g ( x ) =$ __________, then $( g \circ f ) ( x ) = 3 x - 1$

Accepted Answer

To find the function $g(x)$ such that $(g \circ f)(x) = 3x - 1$, we need to understand that the composition of functions $(g \circ f)(x)$ means applying the function $f$ to $x$ first, and then applying the function $g$ to the result of $f(x)$.

Given $f(x) = \sqrt{x}$, we want to find $g(x)$ such that when we apply $g$ to $f(x)$, we get $3x - 1$.

Let's denote $y = f(x) = \sqrt{x}$. Then, we want to find $g(y)$ such that $g(y) = 3x - 1$.

Since $y = \sqrt{x}$, we can square both sides to find $x$:

$y^2 = (\sqrt{x})^2$
$y^2 = x$

Now, we substitute $y^2$ for $x$ in the equation $g(y) = 3x - 1$:

$g(y) = 3y^2 - 1$

Therefore, the function $g(x)$ that satisfies the condition $(g \circ f)(x) = 3x - 1$ is:

$g(x) = 3x^2 - 1$

Question 3

If $f ( x ) = x ^ { 3 } \text { and } g ( x ) =$ __________, then $( f \circ g ) ( x ) = x ^ { 3 } + 9 x ^ { 2 } + 27 x + 27$

Accepted Answer

To find the function $g(x)$ such that $(f \circ g)(x) = x^3 + 9x^2 + 27x + 27$, we need to determine what function $g$ when composed with $f(x) = x^3$ results in the given polynomial.

Given that $f(x) = x^3$, the composition $(f \circ g)(x)$ means we are taking the output of $g(x)$ and plugging it into $f$. In other words, we are cubing whatever $g(x)$ is. So, we are looking for a function $g(x)$ such that when we cube it, we get $x^3 + 9x^2 + 27x + 27$.

Let's assume $g(x) = ax^2 + bx + c$. Then we have:

$$
f(g(x)) = (ax^2 + bx + c)^3
$$

We need to expand this and match it to the given polynomial $x^3 + 9x^2 + 27x + 27$. However, we can notice that the given polynomial can be factored as:

$$
x^3 + 9x^2 + 27x + 27 = (x + 3)^3
$$

This suggests that $g(x)$ should be $x + 3$, because when we plug $x + 3$ into $f(x) = x^3$, we get:

$$
f(g(x)) = f(x + 3) = (x + 3)^3 = x^3 + 9x^2 + 27x + 27
$$

Therefore, the function $g(x)$ that satisfies the given condition is:

$$
g(x) = x + 3
$$

Question 4

Let $f ( x ) = 7 x - 6 \text { and } g ( x ) = - 2 x + 2$ . Find: $f + g$
A) $- 14 x ^ { 2 } + 26 x - 12$
B) $5 x + 4$
C) $5 x - 4$
D) $5 x - 8$

Accepted Answer

The answer of Let \(f ( x ) = 7...

Question 5

Let $f ( x ) = 3 x - 4$ and $g ( x ) = x ^ { 2 } - 1$ . Find: $( g \circ f ) ( - 1 )$
A) -4
B) 0
C) $- 3 x ^ { 2 } + 24 x - 15$
D) 48

Accepted Answer

The answer of Let \(f ( x ) = 3...

Question 6

Let $f ( x ) = 2 x + 1$ and $g ( x ) = x ^ { 2 } - 4$ . Find the composition. $( g \circ f ) ( 2 )$

Accepted Answer

The answer of Let \(f ( x ) = 2...

Question 7

Let $f ( x ) = 5 x - 5 \text { and } g ( x ) = - 2 x + 9$ . Find: $( f - g ) ( 3 )$ .
A) 13
B) 7
C) 20
D) 30

Accepted Answer

The answer of Let \(f ( x ) = 5...

Question 8

Let $f ( x ) = 5 x - 4$ and $g ( x ) = x ^ { 2 } - 6$ . Find: f g
A) $- x ^ { 2 } + 5 x + 2$
B) $- x ^ { 2 } + 5 x - 10$
C) $x ^ { 2 } - 5 x - 2$
D) $- x ^ { 2 } + 5 x - 2$

Accepted Answer

The answer of Let \(f ( x ) = 5...

Question 9

The __________ of the function f+g is the set of real numbers x that are in the domain of both f and g .

Accepted Answer

The answer of The __________ of the function f+g is...

Question 10

Let $f ( x ) = 5 x - 9$ and $g ( x ) = 4 x - 2$ . Find the domain of $\frac { f } { g }$ .
A) $( - \infty , 0 ) \cup ( 0 , \infty )$
B) $( - \infty , \infty )$
C) $\left( - \infty , \frac { 9 } { 5 } \right) \cup \left( \frac { 9 } { 5 } , \infty \right)$
D) $\left( - \infty , \frac { 1 } { 2 } \right) \cup \left( \frac { 1 } { 2 } , \infty \right)$

Accepted Answer

The domain of $\frac{f}{g}$ is all real numbers except where $g(x) = 0$. Solving $4x - 2 = 0$ gives $x = \frac{1}{2}$, so the domain excludes $\frac{1}{2}$, leading to the answer $\left( - \infty , \frac { 1 } { 2 } \right) \cup \left( \frac { 1 } { 2 } , \infty \right)$.

Question 11

Let $f ( x ) = 5 x + 4$ and $g ( x ) = x ^ { 2 } - 2$ . Find the composition. $( f \circ g ) ( 3 )$

Accepted Answer

The answer of Let \(f ( x ) = 5...

Question 12

Let $f ( x ) = 3 x - 5$ and $g ( x ) = x ^ { 2 } - 3$ . Find:  f . g 
A) $3 x ^ { 2 } - 15$
B) $3 x ^ { 3 } + 15$
C) $3 x ^ { 3 } - 5 x ^ { 2 } - 9 x + 15$
D) $3 x ^ { 3 } - 5 x ^ { 2 } - 9 x - 15$

Accepted Answer

The answer of Let \(f ( x ) = 3...

Question 13

Let $f ( x ) = 4 x - 7 \text { and } g ( x ) = - 2 x + 9$ Find: $( f \cdot g ) ( 1 )$
A) 21
B) -21
C) -16
D) 0

Accepted Answer

To find $(f \cdot g)(1)$, first evaluate $f(1)$ and $g(1)$, then multiply the results. $f(1) = 4(1) - 7 = -3$ and $g(1) = -2(1) + 9 = 7$. Multiplying these gives $-3 \cdot 7 = -21$.

Question 14

Let $f ( x ) = 4 x - 3$ and $g ( x ) = x ^ { 2 } + x$ . Find the composition. $( g \circ f ) ( - 3 )$

Accepted Answer

The answer of Let \(f ( x ) = 4...

Question 15

Let $f ( x ) = 6 x - 5$ and $g ( x ) = x ^ { 2 } + x$ . Find the composition. $( f \circ g ) ( - 3 )$

Accepted Answer

The answer of Let \(f ( x ) = 6...

Question 16

Let $f ( x ) = 4 x + 3$ and $g ( x ) = x ^ { 2 } - 1$. Find the composition. $( f \circ g ) ( x )$

Accepted Answer

The answer of Let \(f ( x ) = 4...

Question 17

The __________ function $g \circ h$ is defined by $( g \circ h ) ( x ) =$ __________.

Accepted Answer

The answer of The __________ function $g \circ h$ is...

Question 18

The __________ of f  and g , denoted as f+g , is defined by $( f + g ) ( x ) =$ __________ and the __________ of $f$ and $g$ , denoted as $f - g$ , is defined by $( f - g ) ( x ) =$ __________.

Accepted Answer

The answer of The __________ of f  and g...

Question 19

The __________ of f and h , denoted as $f \cdot h$ , is defined by $( f \cdot h ) ( x ) =$ __________ and the __________ of $f$ and $h$ , denoted as $\frac { f } { h }$ , is defined by $\left( \frac { f } { h } \right) ( x ) =$ __________.

Accepted Answer

The answer of The __________ of f and h ,...

Question 20

Let $f ( x ) = 5 x + 4$ and $g ( x ) = x ^ { 2 } - 3$ . Find the composition. $( g \circ f ) ( x )$

Accepted Answer

The answer of Let \(f ( x ) = 5...

Quiz 9: Exponential and Logarithmic Functions

Let $f ( x ) = 3 x - 4$ and $g ( x ) = x ^ { 2 } - 1$ . Find: $( f \circ g ) ( x )$

If $f ( x ) = \sqrt { x }$ and $g ( x ) =$ __________, then $( g \circ f ) ( x ) = 3 x - 1$

If $f ( x ) = x ^ { 3 } \text { and } g ( x ) =$ __________, then $( f \circ g ) ( x ) = x ^ { 3 } + 9 x ^ { 2 } + 27 x + 27$

Let $f ( x ) = 7 x - 6 \text { and } g ( x ) = - 2 x + 2$ . Find: $f + g$

Let $f ( x ) = 3 x - 4$ and $g ( x ) = x ^ { 2 } - 1$ . Find: $( g \circ f ) ( - 1 )$

Let $f ( x ) = 2 x + 1$ and $g ( x ) = x ^ { 2 } - 4$ . Find the composition. $( g \circ f ) ( 2 )$

Let $f ( x ) = 5 x - 5 \text { and } g ( x ) = - 2 x + 9$ . Find: $( f - g ) ( 3 )$ .

Let $f ( x ) = 5 x - 4$ and $g ( x ) = x ^ { 2 } - 6$ . Find: f g

The __________ of the function f+g is the set of real numbers x that are in the domain of both f and g .

Let $f ( x ) = 5 x - 9$ and $g ( x ) = 4 x - 2$ . Find the domain of $\frac { f } { g }$ .

Let $f ( x ) = 5 x + 4$ and $g ( x ) = x ^ { 2 } - 2$ . Find the composition. $( f \circ g ) ( 3 )$

Let $f ( x ) = 3 x - 5$ and $g ( x ) = x ^ { 2 } - 3$ . Find: f . g

Let $f ( x ) = 4 x - 7 \text { and } g ( x ) = - 2 x + 9$ Find: $( f \cdot g ) ( 1 )$

Let $f ( x ) = 4 x - 3$ and $g ( x ) = x ^ { 2 } + x$ . Find the composition. $( g \circ f ) ( - 3 )$

Let $f ( x ) = 6 x - 5$ and $g ( x ) = x ^ { 2 } + x$ . Find the composition. $( f \circ g ) ( - 3 )$

Let $f ( x ) = 4 x + 3$ and $g ( x ) = x ^ { 2 } - 1$ . Find the composition. $( f \circ g ) ( x )$

The function $g \circ h$ is defined by $( g \circ h ) ( x ) =$ .

The of f and g , denoted as f+g , is defined by $( f + g ) ( x ) =$ and the of $f$ and $g$ , denoted as $f - g$ , is defined by $( f - g ) ( x ) =$ .

The of f and h , denoted as $f \cdot h$ , is defined by $( f \cdot h ) ( x ) =$ and the of $f$ and $h$ , denoted as $\frac { f } { h }$ , is defined by $\left( \frac { f } { h } \right) ( x ) =$ .

Let $f ( x ) = 5 x + 4$ and $g ( x ) = x ^ { 2 } - 3$ . Find the composition. $( g \circ f ) ( x )$

Quiz 9: Exponential and Logarithmic Functions

Let f(x) =3x−4f ( x ) = 3 x - 4f(x) =3x−4 and g(x) =x2−1g ( x ) = x ^ { 2 } - 1g(x) =x2−1 . Find: (f∘g) (x) ( f \circ g ) ( x ) (f∘g) (x)

If f(x)=xf ( x ) = \sqrt { x }f(x)=x​ and g(x)=g ( x ) =g(x)= __________, then (g∘f)(x)=3x−1( g \circ f ) ( x ) = 3 x - 1(g∘f)(x)=3x−1

If f(x)=x3 and g(x)=f ( x ) = x ^ { 3 } \text { and } g ( x ) =f(x)=x3 and g(x)= __________, then (f∘g)(x)=x3+9x2+27x+27( f \circ g ) ( x ) = x ^ { 3 } + 9 x ^ { 2 } + 27 x + 27(f∘g)(x)=x3+9x2+27x+27

Let f(x) =7x−6 and g(x) =−2x+2f ( x ) = 7 x - 6 \text { and } g ( x ) = - 2 x + 2f(x) =7x−6 and g(x) =−2x+2 . Find: f+gf + gf+g

Let f(x) =3x−4f ( x ) = 3 x - 4f(x) =3x−4 and g(x) =x2−1g ( x ) = x ^ { 2 } - 1g(x) =x2−1 . Find: (g∘f) (−1) ( g \circ f ) ( - 1 ) (g∘f) (−1)

Let f(x)=2x+1f ( x ) = 2 x + 1f(x)=2x+1 and g(x)=x2−4g ( x ) = x ^ { 2 } - 4g(x)=x2−4 . Find the composition. (g∘f)(2)( g \circ f ) ( 2 )(g∘f)(2)

Let f(x) =5x−5 and g(x) =−2x+9f ( x ) = 5 x - 5 \text { and } g ( x ) = - 2 x + 9f(x) =5x−5 and g(x) =−2x+9 . Find: (f−g) (3) ( f - g ) ( 3 ) (f−g) (3) .

Let f(x) =5x−4f ( x ) = 5 x - 4f(x) =5x−4 and g(x) =x2−6g ( x ) = x ^ { 2 } - 6g(x) =x2−6 . Find: f g

The __________ of the function f+g is the set of real numbers x that are in the domain of both f and g .

Let f(x) =5x−9f ( x ) = 5 x - 9f(x) =5x−9 and g(x) =4x−2g ( x ) = 4 x - 2g(x) =4x−2 . Find the domain of fg\frac { f } { g }gf​ .

Let f(x)=5x+4f ( x ) = 5 x + 4f(x)=5x+4 and g(x)=x2−2g ( x ) = x ^ { 2 } - 2g(x)=x2−2 . Find the composition. (f∘g)(3)( f \circ g ) ( 3 )(f∘g)(3)

Let f(x) =3x−5f ( x ) = 3 x - 5f(x) =3x−5 and g(x) =x2−3g ( x ) = x ^ { 2 } - 3g(x) =x2−3 . Find: f . g

Let f(x) =4x−7 and g(x) =−2x+9f ( x ) = 4 x - 7 \text { and } g ( x ) = - 2 x + 9f(x) =4x−7 and g(x) =−2x+9 Find: (f⋅g) (1) ( f \cdot g ) ( 1 ) (f⋅g) (1)

Let f(x)=4x−3f ( x ) = 4 x - 3f(x)=4x−3 and g(x)=x2+xg ( x ) = x ^ { 2 } + xg(x)=x2+x . Find the composition. (g∘f)(−3)( g \circ f ) ( - 3 )(g∘f)(−3)

Let f(x)=6x−5f ( x ) = 6 x - 5f(x)=6x−5 and g(x)=x2+xg ( x ) = x ^ { 2 } + xg(x)=x2+x . Find the composition. (f∘g)(−3)( f \circ g ) ( - 3 )(f∘g)(−3)

Let f(x)=4x+3f ( x ) = 4 x + 3f(x)=4x+3 and g(x)=x2−1g ( x ) = x ^ { 2 } - 1g(x)=x2−1 . Find the composition. (f∘g)(x)( f \circ g ) ( x )(f∘g)(x)

The __________ function g∘hg \circ hg∘h is defined by (g∘h)(x)=( g \circ h ) ( x ) =(g∘h)(x)= __________.

The __________ of f and g , denoted as f+g , is defined by (f+g)(x)=( f + g ) ( x ) =(f+g)(x)= __________ and the __________ of fff and ggg , denoted as f−gf - gf−g , is defined by (f−g)(x)=( f - g ) ( x ) =(f−g)(x)= __________.

The __________ of f and h , denoted as f⋅hf \cdot hf⋅h , is defined by (f⋅h)(x)=( f \cdot h ) ( x ) =(f⋅h)(x)= __________ and the __________ of fff and hhh , denoted as fh\frac { f } { h }hf​ , is defined by (fh)(x)=\left( \frac { f } { h } \right) ( x ) =(hf​)(x)= __________.

Let f(x)=5x+4f ( x ) = 5 x + 4f(x)=5x+4 and g(x)=x2−3g ( x ) = x ^ { 2 } - 3g(x)=x2−3 . Find the composition. (g∘f)(x)( g \circ f ) ( x )(g∘f)(x)

Let $f ( x ) = 3 x - 4$ and $g ( x ) = x ^ { 2 } - 1$ . Find: $( f \circ g ) ( x )$

If $f ( x ) = \sqrt { x }$ and $g ( x ) =$ __________, then $( g \circ f ) ( x ) = 3 x - 1$

If $f ( x ) = x ^ { 3 } \text { and } g ( x ) =$ __________, then $( f \circ g ) ( x ) = x ^ { 3 } + 9 x ^ { 2 } + 27 x + 27$

Let $f ( x ) = 7 x - 6 \text { and } g ( x ) = - 2 x + 2$ . Find: $f + g$

Let $f ( x ) = 3 x - 4$ and $g ( x ) = x ^ { 2 } - 1$ . Find: $( g \circ f ) ( - 1 )$

Let $f ( x ) = 2 x + 1$ and $g ( x ) = x ^ { 2 } - 4$ . Find the composition. $( g \circ f ) ( 2 )$

Let $f ( x ) = 5 x - 5 \text { and } g ( x ) = - 2 x + 9$ . Find: $( f - g ) ( 3 )$ .

Let $f ( x ) = 5 x - 4$ and $g ( x ) = x ^ { 2 } - 6$ . Find: f g

Let $f ( x ) = 5 x - 9$ and $g ( x ) = 4 x - 2$ . Find the domain of $\frac { f } { g }$ .

Let $f ( x ) = 5 x + 4$ and $g ( x ) = x ^ { 2 } - 2$ . Find the composition. $( f \circ g ) ( 3 )$

Let $f ( x ) = 3 x - 5$ and $g ( x ) = x ^ { 2 } - 3$ . Find: f . g

Let $f ( x ) = 4 x - 7 \text { and } g ( x ) = - 2 x + 9$ Find: $( f \cdot g ) ( 1 )$

Let $f ( x ) = 4 x - 3$ and $g ( x ) = x ^ { 2 } + x$ . Find the composition. $( g \circ f ) ( - 3 )$

Let $f ( x ) = 6 x - 5$ and $g ( x ) = x ^ { 2 } + x$ . Find the composition. $( f \circ g ) ( - 3 )$

Let $f ( x ) = 4 x + 3$ and $g ( x ) = x ^ { 2 } - 1$ . Find the composition. $( f \circ g ) ( x )$

The function $g \circ h$ is defined by $( g \circ h ) ( x ) =$ .

The of f and g , denoted as f+g , is defined by $( f + g ) ( x ) =$ and the of $f$ and $g$ , denoted as $f - g$ , is defined by $( f - g ) ( x ) =$ .

The of f and h , denoted as $f \cdot h$ , is defined by $( f \cdot h ) ( x ) =$ and the of $f$ and $h$ , denoted as $\frac { f } { h }$ , is defined by $\left( \frac { f } { h } \right) ( x ) =$ .

Let $f ( x ) = 5 x + 4$ and $g ( x ) = x ^ { 2 } - 3$ . Find the composition. $( g \circ f ) ( x )$