Question 1

Which of the following points lie on the graph of $$f ( x ) = \sqrt { 3 x + 4 }$$ ?A (-3, 6) B (4, 4) C (0, 4)
A) A and B
B) B and C
C) Only B
D) Only C

Accepted Answer

Only point B (4, 4) satisfies the equation $f(x) = \sqrt{3x + 4}$ when plugged in: $f(4) = \sqrt{3(4) + 4} = \sqrt{12 + 4} = \sqrt{16} = 4$. Points A and C do not satisfy the equation.

Question 2

Describe the function $f ( t ) = - 5 t + 1$ as a set of ordered pairs.

Accepted Answer

$( t , 5 t + 1 )$

Question 3

Construct a rule $f : \{ 1,2,3,4,5 \} \rightarrow \{ 3,6,9,12 \}$ that is (a) a function,(b) a one-to-one function.

Accepted Answer

(a) $$f ( 1 ) = f ( 2 ) = 3 , f ( 3 ) = 6 , f ( 4 ) = 9 , \text { and } f ( 5 ) = 12$$ (b) $$f ( 1 ) = 3 , f ( 3 ) = 6 , f ( 4 ) = 9 , \text { and } f ( 5 ) = 12$$ 
 (There are other possible answers.)

Question 4

Find the domain of $f ( x ) = \frac { \sqrt { x ^ { 2 } - 4 } } { x ^ { 2 } - 9 }$

Accepted Answer

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

Question 5

Find the average rate of change of the function $f ( x ) = \sqrt { x + 1 }$ on the interval [0, 8].
A) 8/3
B) 3/8
C) 1/4
D) -1/4

Accepted Answer

The answer of Find the average rate of change of...

Question 6

Find the domain and range of $f ( x ) = \sqrt { 2 - x } + 1$

Accepted Answer

The answer of Find the domain and range of \(f...

Question 7

Find the domain and range of $f ( x ) = \frac { 1 } { \sqrt { x - 3 } }$
A) Domain: $( - \infty , 3 ]$ Range: $[ 0 , \infty )$
B) Domain: $[ 0 , \infty )$ Range: $( - \infty , 3 ]$
C) Domain: $( - \infty , 3 )$ Range: $( 0 , \infty )$
D) Domain: $( 3 , \infty )$ Range: $( 0 , \infty )$

Accepted Answer

The function $f(x) = \frac{1}{\sqrt{x-3}}$ is defined for $x > 3$ because the square root requires a non-negative argument, and the denominator cannot be zero. The range is $(0, \infty)$ because as $x$ increases, $f(x)$ approaches 0 but never reaches it, and as $x$ approaches 3 from the right, $f(x)$ increases without bound.

Question 8

Find the domain and range of $f ( x ) = \frac { 1 } { x ^ { 2 } + 4 }$
A) Domain: $( - \infty , - 2 ) \cup ( - 2,2 ) \cup ( 2 , \infty )$ Range: $( - \infty , \infty )$
B) Domain: $( - \infty , \infty )$ Range: $( - \infty , \infty )$
C) Domain: $( - \infty , \infty )$ Range: $( 0 , \infty )$
D) Domain: $( - \infty , - 2 ) \cup ( - 2,2 ) \cup ( 2 , \infty )$ Range: $[ 0 , \infty )$

Accepted Answer

The denominator of the function is never zero, so the domain is all real numbers. The range is all positive numbers because the square of any real number is always positive, and adding 4 to that will always result in a positive number.

Question 9

Find the domain and range of $f ( x ) = \frac { | x - 2 | } { x - 2 }$
A) Domain: $( - \infty , - 2 ) \cup ( - 2,2 ) \cup ( 2 , \infty )$ Range: $( - \infty , \infty )$
B) Domain: $( - \infty , \infty )$ Range: $( - \infty  , \infty )$
C) Domain: $( - \infty , - 2 ) \cup ( - 2,2 ) \cup (  2 , \infty )$ Range: $( - \infty , \infty )$
D) Domain: $( - \infty , - 2 ) \cup ( - 2,2 ) \cup ( 2  , \infty )$ Range: $\{ - 1,1 \}$

Accepted Answer

The function is undefined at $x = 2$ because the denominator becomes zero, so the domain excludes 2. The function outputs either 1 or -1 depending on the sign of $x - 2$, hence the range is \{-1, 1\}.

Question 10

Find the domain and range of $f ( x ) = \frac { \sqrt { 9 - x ^ { 2 } } } { x + 3 }$
A) Domain: [-3, 3] Range: $[ 0 , \infty )$
B) Domain: (-3, 3) Range: $[ 0 , \infty )$
C) Domain: (-3, 3] Range: $[ 0 , \infty )$
D) Domain: (-3, 3] Range: $( 0 , \infty )$

Accepted Answer

The answer of Find the domain and range of \(f...

Question 11

Find the domain and range of $f ( x ) = \sqrt { x ^ { 2 } - 5 x + 4 }$
A) Domain: $( - \infty , - 4 ) \cup ( - 1 , \infty )$ Range: $( 0 , \infty )$
B) Domain: $( - \infty , 1 ) \cup ( 4 , \infty )$ Range: $[ 0 , \infty )$
C) Domain: $( - \infty , 1 ] \cup [ 4 , \infty )$ Range: $[ 0 , \infty )$
D) Domain: $( - \infty , 1 ] \cup [ 4 , \infty )$ Range: $( 0 , \infty )$

Accepted Answer

The expression inside the square root, $x^2 - 5x + 4$, must be greater than or equal to 0 for the square root to be real. Factoring the quadratic gives $(x-1)(x-4) \geq 0$, which is true for $x \leq 1$ or $x \geq 4$. Thus, the domain is $(-\infty, 1] \cup [4, \infty)$. The range of a square root function is always $[0, \infty)$ because the output of a square root is never negative.

Question 12

If $f ( x ) = x ^ { 3 } + 2$ , find $f \left( a ^ { 2 } \right)$
A) $a ^ { 5 } + 2$
B) $\left( a ^ { 2 } + 2 \right) ^ { 3 }$
C) $a ^ { 3 } + 8$
D) $a ^ { 6 } + 2$

Accepted Answer

To find $f(a^2)$, substitute $a^2$ for $x$ in the given function: $f(a^2) = (a^2)^3 + 2 = a^{6} + 2$.

Question 13

Given $y = 1 - \sqrt { x }$ , for what value(s) of $X$ is $y = - 2$ ?
A) 4
B) 9
C) 16
D) None of these

Accepted Answer

The answer of Given \(y = 1 - \sqrt {...

Question 14

Find the domain of $f ( x ) = \frac { x ^ { 2 } - 9 } { \sqrt { x + 3 } }$
A) Domain: $[ - 3 , \infty )$
B) Domain: $( - 3 , \infty )$
C) Domain: $( - \infty , \infty )$
D) Domain: $( - \infty , - 3 ) \cup ( - 3 , \infty )$

Accepted Answer

The domain of $f(x) = \frac{x^2 - 9}{\sqrt{x + 3}}$ is determined by the conditions under which the expression is defined. The denominator, $\sqrt{x + 3}$, requires $x + 3 > 0$ (since the square root of a negative number is not real), which simplifies to $x > -3$. Therefore, the domain is all real numbers greater than -3, which is represented by the interval $(-3, \infty)$.

Question 15

Given $y = x ^ { 2 } - 4 x - 8$ , for what value(s) of $X$ is $y = - 3$ ?
A) -5 and 1
B) 5 and 1
C) 4 and 2
D) -1 and 5

Accepted Answer

The answer of Given \(y = x ^ { 2...

Question 16

If $f ( x ) = \sqrt { 2 x - 1 }$ , find $\frac { f ( 1 + h ) - f ( 1 ) } { h }$

Accepted Answer

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

Question 17

If $g ( x ) = \frac { x ^ { 2 } + 1 } { x }$ , find $g ( x + h ) - g ( x )$

Accepted Answer

The answer of If \(g ( x ) = \frac...

Question 18

Which of the following is an odd function?
A) $f ( x ) = x ^ { 3 } - x ^ { 2 }$
B) $g ( x ) = \frac { 5 } { x }$
C) $h ( x ) = x ^ { 4 } - x ^ { 2 }$
D) $k ( x ) = | x - 2 |$

Accepted Answer

An odd function satisfies the condition $f(-x) = -f(x)$. For choice B, $g(-x) = \frac{5}{-x} = -\frac{5}{x} = -g(x)$, which makes it an odd function. Choices A, C, and D do not satisfy this condition.

Question 19

Is $f ( x ) = \frac { 2 x ^ { 3 } } { x ^ { 2 } + 2 }$ an even function, an odd function or neither? Explain.

Accepted Answer

The answer of Is \(f ( x ) = \frac...

Question 20

Is $g ( x ) = \frac { - 3 } { x ^ { 2 } + 1 }$ an even function, an odd function or neither? Explain.

Accepted Answer

The answer of Is \(g ( x ) = \frac...

Quiz 15: Functions and Precalculus

Which of the following points lie on the graph of $f ( x ) = \sqrt { 3 x + 4 }$ ?A (-3, 6) B (4, 4) C (0, 4)

Describe the function $f ( t ) = - 5 t + 1$ as a set of ordered pairs.

Construct a rule $f : \{ 1,2,3,4,5 \} \rightarrow \{ 3,6,9,12 \}$ that is (a) a function,(b) a one-to-one function.

Find the domain of $f ( x ) = \frac { \sqrt { x ^ { 2 } - 4 } } { x ^ { 2 } - 9 }$

Find the average rate of change of the function $f ( x ) = \sqrt { x + 1 }$ on the interval [0, 8].

Find the domain and range of $f ( x ) = \sqrt { 2 - x } + 1$

Find the domain and range of $f ( x ) = \frac { 1 } { \sqrt { x - 3 } }$

Find the domain and range of $f ( x ) = \frac { 1 } { x ^ { 2 } + 4 }$

Find the domain and range of $f ( x ) = \frac { | x - 2 | } { x - 2 }$

Find the domain and range of $f ( x ) = \frac { \sqrt { 9 - x ^ { 2 } } } { x + 3 }$

Find the domain and range of $f ( x ) = \sqrt { x ^ { 2 } - 5 x + 4 }$

If $f ( x ) = x ^ { 3 } + 2$ , find $f \left( a ^ { 2 } \right)$

Given $y = 1 - \sqrt { x }$ , for what value(s) of $X$ is $y = - 2$ ?

Find the domain of $f ( x ) = \frac { x ^ { 2 } - 9 } { \sqrt { x + 3 } }$

Given $y = x ^ { 2 } - 4 x - 8$ , for what value(s) of $X$ is $y = - 3$ ?

If $f ( x ) = \sqrt { 2 x - 1 }$ , find $\frac { f ( 1 + h ) - f ( 1 ) } { h }$

If $g ( x ) = \frac { x ^ { 2 } + 1 } { x }$ , find $g ( x + h ) - g ( x )$

Which of the following is an odd function?

Is $f ( x ) = \frac { 2 x ^ { 3 } } { x ^ { 2 } + 2 }$ an even function, an odd function or neither? Explain.

Is $g ( x ) = \frac { - 3 } { x ^ { 2 } + 1 }$ an even function, an odd function or neither? Explain.

Quiz 15: Functions and Precalculus

Which of the following points lie on the graph of f(x) =3x+4f ( x ) = \sqrt { 3 x + 4 }f(x) =3x+4​ ?A (-3, 6) B (4, 4) C (0, 4)

Describe the function f(t)=−5t+1f ( t ) = - 5 t + 1f(t)=−5t+1 as a set of ordered pairs.

Construct a rule f:{1,2,3,4,5}→{3,6,9,12}f : \{ 1,2,3,4,5 \} \rightarrow \{ 3,6,9,12 \}f:{1,2,3,4,5}→{3,6,9,12} that is (a) a function,(b) a one-to-one function.

Find the domain of f(x)=x2−4x2−9f ( x ) = \frac { \sqrt { x ^ { 2 } - 4 } } { x ^ { 2 } - 9 }f(x)=x2−9x2−4​​

Find the average rate of change of the function f(x) =x+1f ( x ) = \sqrt { x + 1 }f(x) =x+1​ on the interval [0, 8].

Find the domain and range of f(x)=2−x+1f ( x ) = \sqrt { 2 - x } + 1f(x)=2−x​+1

Find the domain and range of f(x) =1x−3f ( x ) = \frac { 1 } { \sqrt { x - 3 } }f(x) =x−3​1​

Find the domain and range of f(x) =1x2+4f ( x ) = \frac { 1 } { x ^ { 2 } + 4 }f(x) =x2+41​

Find the domain and range of f(x) =∣x−2∣x−2f ( x ) = \frac { | x - 2 | } { x - 2 }f(x) =x−2∣x−2∣​

Find the domain and range of f(x) =9−x2x+3f ( x ) = \frac { \sqrt { 9 - x ^ { 2 } } } { x + 3 }f(x) =x+39−x2​​

Find the domain and range of f(x) =x2−5x+4f ( x ) = \sqrt { x ^ { 2 } - 5 x + 4 }f(x) =x2−5x+4​

If f(x) =x3+2f ( x ) = x ^ { 3 } + 2f(x) =x3+2 , find f(a2) f \left( a ^ { 2 } \right) f(a2)

Given y=1−xy = 1 - \sqrt { x }y=1−x​ , for what value(s) of XXX is y=−2y = - 2y=−2 ?

Find the domain of f(x) =x2−9x+3f ( x ) = \frac { x ^ { 2 } - 9 } { \sqrt { x + 3 } }f(x) =x+3​x2−9​

Given y=x2−4x−8y = x ^ { 2 } - 4 x - 8y=x2−4x−8 , for what value(s) of XXX is y=−3y = - 3y=−3 ?

If f(x)=2x−1f ( x ) = \sqrt { 2 x - 1 }f(x)=2x−1​ , find f(1+h)−f(1)h\frac { f ( 1 + h ) - f ( 1 ) } { h }hf(1+h)−f(1)​

If g(x)=x2+1xg ( x ) = \frac { x ^ { 2 } + 1 } { x }g(x)=xx2+1​ , find g(x+h)−g(x)g ( x + h ) - g ( x )g(x+h)−g(x)

Which of the following is an odd function?

Is f(x)=2x3x2+2f ( x ) = \frac { 2 x ^ { 3 } } { x ^ { 2 } + 2 }f(x)=x2+22x3​ an even function, an odd function or neither? Explain.

Is g(x)=−3x2+1g ( x ) = \frac { - 3 } { x ^ { 2 } + 1 }g(x)=x2+1−3​ an even function, an odd function or neither? Explain.

Which of the following points lie on the graph of $f ( x ) = \sqrt { 3 x + 4 }$ ?A (-3, 6) B (4, 4) C (0, 4)

Describe the function $f ( t ) = - 5 t + 1$ as a set of ordered pairs.

Construct a rule $f : \{ 1,2,3,4,5 \} \rightarrow \{ 3,6,9,12 \}$ that is (a) a function,(b) a one-to-one function.

Find the domain of $f ( x ) = \frac { \sqrt { x ^ { 2 } - 4 } } { x ^ { 2 } - 9 }$

Find the average rate of change of the function $f ( x ) = \sqrt { x + 1 }$ on the interval [0, 8].

Find the domain and range of $f ( x ) = \sqrt { 2 - x } + 1$

Find the domain and range of $f ( x ) = \frac { 1 } { \sqrt { x - 3 } }$

Find the domain and range of $f ( x ) = \frac { 1 } { x ^ { 2 } + 4 }$

Find the domain and range of $f ( x ) = \frac { | x - 2 | } { x - 2 }$

Find the domain and range of $f ( x ) = \frac { \sqrt { 9 - x ^ { 2 } } } { x + 3 }$

Find the domain and range of $f ( x ) = \sqrt { x ^ { 2 } - 5 x + 4 }$

If $f ( x ) = x ^ { 3 } + 2$ , find $f \left( a ^ { 2 } \right)$

Given $y = 1 - \sqrt { x }$ , for what value(s) of $X$ is $y = - 2$ ?

Find the domain of $f ( x ) = \frac { x ^ { 2 } - 9 } { \sqrt { x + 3 } }$

Given $y = x ^ { 2 } - 4 x - 8$ , for what value(s) of $X$ is $y = - 3$ ?

If $f ( x ) = \sqrt { 2 x - 1 }$ , find $\frac { f ( 1 + h ) - f ( 1 ) } { h }$

If $g ( x ) = \frac { x ^ { 2 } + 1 } { x }$ , find $g ( x + h ) - g ( x )$

Is $f ( x ) = \frac { 2 x ^ { 3 } } { x ^ { 2 } + 2 }$ an even function, an odd function or neither? Explain.

Is $g ( x ) = \frac { - 3 } { x ^ { 2 } + 1 }$ an even function, an odd function or neither? Explain.