Question 1

Determine whether or not the graph is a graph of a function of x.
-
A) Function
B) Not a function

Accepted Answer

The graph is not a function of x because there are multiple values of y for the same value of x. This occurs in the middle section of the graph, where there are two horizontal lines at different heights intersecting the same vertical line.

Question 2

Find the formula for the function.
-Express the area of a circle as a function of its radius r. 
A) $A = 2 \pi r$
B) $A = \pi r ^ { 2 }$
C) $\mathrm { A } = \pi \mathrm { r }$
D) $A = \pi r ^ { 3 }$

Accepted Answer

The formula for the area of a circle is A = πr^2. This is the only choice that correctly represents this formula. Choices A, C, and D all have incorrect exponents or do not include π.

Question 3

Find the domain and range of the function.
-$$f ( x ) = 4 - x ^ { 2 }$$
A) D: $( - \infty , \infty ) , R : ( - \infty , 4 ]$
B) D: $( - \infty , \infty ) , R : ( - \infty , \infty )$
C) D: $( - \infty , 4 ] , R : ( - \infty , \infty )$
D) D: $( - \infty , \infty )$, R: $[ 4 , \infty )$

Accepted Answer

The domain of this function is all real numbers since there are no restrictions on the input value of x.
The range of the function can be found by looking at the highest and lowest possible outputs. The highest output is 4, which occurs when x = 0, and the lowest output is negative infinity since the function can approach negative infinity as x becomes very large in either the positive or negative direction. Therefore, the range is ( - infinity, 4 ].

Question 4

Determine whether or not the graph is a graph of a function of x.
-
A) Function
B) Not a function

Accepted Answer

The graph has multiple vertical lines passing through it, indicating that there are multiple y-values for the same x-value. Therefore, it is not a function of x.

Question 5

Find the formula for the function.
-Express the perimeter of a square as a function of the square's side length x. 
A) $p = \frac { 3 x } { 2 }$
B) $p = x ^ { 3 }$
C) $p = 6 x$
D) $p = 4 x$

Accepted Answer

The perimeter of a square is the sum of all its sides, which is equal to 4 times its side length. Therefore, the formula for the function is p = 4x. This is option D.

Question 6

Find the formula for the function.
-Express the length d of a square's diagonal as a function of its side length x. 
A) $d = 2 x$
B) $d = x \sqrt { 3 }$
C) $d = x$
D) $d = x \sqrt { 2 }$

Accepted Answer

In a square, the diagonal, side length, and hypotenuse of the right triangle formed by the diagonal and two sides are related by the Pythagorean Theorem:

$d^2 = x^2 + x^2 = 2x^2$

To solve for d, take the square root of both sides:

$d = \sqrt{2x^2} = x\sqrt{2}$.

Therefore, the function for d in terms of x is $d = x\sqrt{2}$, which is answer choice D.

Question 7

Find the domain and range of the function.
-$$f ( x ) = - 2 + \sqrt { x }$$
A) D: $( - \infty , 0 ]$, R: $( - \infty , - 2 ]$
B) D: $[ 0 , \infty ) , R : [ - 2 , \infty )$
C) D: $( - \infty , \infty )$, R: $[ - 2 , \infty )$
D) D: $[ 0 , \infty )$, R: $( - \infty , \infty )$

Accepted Answer

The domain must satisfy two conditions: 
1) The expression under the square root must be nonnegative 
2) The value inside the square root cannot be negative

Solve the inequality $x\geq 0$ to find the domain.

$$x\geq 0$$
$$x^{\frac{1}{2}}\geq 0$$

Therefore, the domain is $[0,\infty)$.

Since $\sqrt{x}$ outputs a nonnegative value, the minimum value of $f(x)$ is $-2$ which occurs at $x=0$.

Therefore, the range of the function is $[-2, \infty)$.

The only answer choice that includes the correct domain and range is B.

Question 8

Find the domain and range of the function.
-$$F ( t ) = t ^ { 2 } - 3$$
A) D: $( - \infty , \infty )$, R: $[ - 3 , \infty )$
B) D: $( - \infty , \infty )$, R: $( - \infty , \infty )$
C) D: $[ 0 , \infty )$, R: $( - \infty , - 3 ]$
D) D: $[ - 9 , \infty )$, R: $[ - 3 , \infty )$

Accepted Answer

Since the function is a polynomial with degree 2, it is defined for all real numbers, thus the domain is $( - \infty , \infty )$. To find the range, we can determine the minimum value of the function by completing the square: 
$t^2 - 3 = (t-0)^2 - 3$. 
Since $(t-0)^2 \geq 0$ for all $t$, the minimum value of the function is $-3$, which is achieved at $t=0$. Therefore, the range is $[-3, \infty)$.

Question 9

Find the domain and range of the function.
-$$f ( x ) = \frac { 7 } { 10 + \sqrt { x } }$$
A) D: $( - \infty , \infty ) , R : \left( 0 , \frac { 7 } { 10 } \right]$
B) D: $[ 0 , \infty ) , \mathrm { R } : \left[ 0 , \frac { 7 } { 10 } \right]$
C) D: $[ 0 , \infty )$, R: $( - \infty , \infty )$
D) D: $( - \infty , 0 ] , \mathrm { R } : ( - \infty , 0 ]$

Accepted Answer

- The function is defined for all non-negative values of x (i.e., x can be 0 or any positive number), so the domain is [0,∞).
- The denominator of the function (10 + √x) can take any non-negative value (since the square root of an non-negative number is non-negative), so the range of the function is also non-negative. 
- Further, as x increases from 0, the denominator of the function increases (because √x increases), so the value of the function decreases. 
- Thus, the largest possible value of the function occurs when x = 0 (i.e., when the denominator is largest), giving us f(0) = 7/10. 
- Therefore, the range of the function is [0,7/10].

Question 10

Find the domain and range of the function.
-$$g ( z ) = - 2 - \sqrt { z }$$
A) D: $( - \infty , 0 ]$, R: $[ - 2 , \infty )$
B) D: $( - \infty , \infty ) , \mathrm { R } : ( - \infty , - 2 ]$
C) D: $[ 0 , \infty ) , \mathrm { R } : ( - \infty , - 2 ]$
D) D: $( - \infty , - 2 ] , R : ( - \infty , \infty )$

Accepted Answer

The square root function $\sqrt{z}$ is defined for $z \geq 0$, so the domain is $[0, \infty)$. The range is determined by the output values, which decrease as $z$ increases, starting from $-2$ (when $z = 0$), hence the range is $(-\infty, -2]$.

Question 11

Find the domain and range of the function.
-$$F ( t ) = \frac { 10 } { 9 \sqrt { t } }$$
A) D: $( - \infty , 0 )$, R: $( - \infty , 0 )$
B) D: $( - \infty , \infty )$, R: $( - \infty , \infty )$
C) D: $[ 0 , \infty ) , R : [ 0 , \infty )$
D) D: $( 0 , \infty ) , R : ( 0 , \infty )$

Accepted Answer

The function involves a square root of t in the denominator, so t cannot be negative or 0. Therefore, the domain is (0, ∞).
For the range, as t approaches infinity, the denominator approaches 0, so the function value becomes very large (in absolute value) and negative. As t approaches 0 from the right, the denominator approaches infinity and the function value approaches positive infinity. Therefore, the range is (0, ∞).

Question 12

Determine whether or not the graph is a graph of a function of x.
-
A) Function
B) Not a function

Accepted Answer

The answer of Determine whether or not the graph is...

Question 13

Determine whether or not the graph is a graph of a function of x.
-
A) Function
B) Not a function

Accepted Answer

The answer of Determine whether or not the graph is...

Question 14

Determine whether or not the graph is a graph of a function of x.
-
A) Function
B) Not a function

Accepted Answer

The answer of Determine whether or not the graph is...

Question 15

Find the formula for the function.
-Express the perimeter of an isosceles triangle with side lengths x, 5x, and 5x as a function of the side length. 
A) $p = 10 x ^ { 3 }$
B) $p = 11 x$
C) $p = 25 x ^ { 3 }$
D) $p = 10 x$

Accepted Answer

The perimeter of a triangle is the sum of its side lengths. For an isosceles triangle with side lengths x, 5x, and 5x, the perimeter $p$ is $x + 5x + 5x = 11x$.

Question 16

Determine whether or not the graph is a graph of a function of x.
-
A) Function
B) Not a function

Accepted Answer

The answer of Determine whether or not the graph is...

Question 17

Find the domain and range of the function.
-$$g ( z ) = \sqrt { 1 - z ^ { 2 } }$$
A) D: $( - 1,1 )$, R: $( - 1,1 )$
B) D: $[ 0 , \infty )$, R: $( - \infty , \infty )$
C) D: $[ - 1,1 ]$, R: $[ 0,1 ]$
D) D: $( - \infty , \infty ) , R : ( 0,1 )$

Accepted Answer

The domain of the function is the set of all values that can be plugged into the function without resulting in an error or undefined output. Here, the function is defined for all values of $z$ that make the expression under the square root non-negative. That is,
$$1-z^2\geq 0 \Rightarrow -1 \leq z\leq 1.$$
Thus, the domain of the function is $[ - 1,1 ]$.
On the other hand, the range of the function is the set of all possible output values. Since the square root function always gives a non-negative output, we have
$$0\leq \sqrt{1-z^2}\leq 1.$$
Therefore, the range of the function is $[0,1]$. Thus, the correct choice is C.

Question 18

Find the formula for the function.
-Express the area of a square as a function of its side length x.
A) A = 2x 
B)$\mathrm{A}=\mathrm{x}^{4}$
C) $A=x^{2}$
D) A = 4x

Accepted Answer

The formula for the area of a square is A = side length^2. Therefore, the correct formula is A = x^2, which is option C.

Question 19

Find the domain and range of the function.
-$$g ( z ) = \frac { - 5 } { \sqrt { z + 1 } }$$
A) D: $( - \infty , - 1 )$, R: $( 0 , \infty )$
B) D: $[ 1 , \infty ) , R : ( - \infty , \infty )$
C) D: $[ 0 , \infty )$, R: $( - \infty , \infty )$
D) D: $( - 1 , \infty ) , \mathrm { R } : ( - \infty , 0 )$

Accepted Answer

The domain of $g(z)$ is all $z$ for which the expression under the square root, $z + 1$, is non-negative. This means $z + 1 > 0$ or $z > -1$, so the domain is $(-1, \infty)$. The range is negative because the numerator is $-5$, and the denominator is the square root of a positive number, making the whole fraction negative. Thus, the range is $(-\infty, 0)$.

Question 20

Find the domain and range of the function.
-$$F ( t ) = \frac { 6 } { \sqrt { t } }$$
A) D: $( - \infty , \infty ) , R : ( - \infty , \infty )$
B) D: $[ 0 , \infty ) , \mathrm { R } : ( - \infty , \infty )$
C) D: $( - \infty , 0 ) , \mathrm { R } : ( - \infty , 0 )$
D) D: $( 0 , \infty ) , \mathrm { R } : ( 0 , \infty )$

Accepted Answer

- The function is defined for all positive values of t since a negative value would give us an imaginary number under the square root. So, the domain is (0, ∞).
- The output of the function is 6 divided by the square root of t. And since the square root of any positive number is also positive, the function will always output positive values. So, the range is also (0, ∞). 
Therefore, the correct choice is D.

Quiz 1: Functions

Determine whether or not the graph is a graph of a function of x. -

Find the formula for the function. -Express the area of a circle as a function of its radius r.

Find the domain and range of the function. - $f ( x ) = 4 - x ^ { 2 }$

Determine whether or not the graph is a graph of a function of x. -

Find the formula for the function. -Express the perimeter of a square as a function of the square's side length x.

Find the formula for the function. -Express the length d of a square's diagonal as a function of its side length x.

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

Find the domain and range of the function. - $F ( t ) = t ^ { 2 } - 3$

Find the domain and range of the function. - $f ( x ) = \frac { 7 } { 10 + \sqrt { x } }$

Find the domain and range of the function. - $g ( z ) = - 2 - \sqrt { z }$

Find the domain and range of the function. - $F ( t ) = \frac { 10 } { 9 \sqrt { t } }$

Determine whether or not the graph is a graph of a function of x. -

Determine whether or not the graph is a graph of a function of x. -

Determine whether or not the graph is a graph of a function of x. -

Find the formula for the function. -Express the perimeter of an isosceles triangle with side lengths x, 5x, and 5x as a function of the side length.

Determine whether or not the graph is a graph of a function of x. -

Find the domain and range of the function. - $g ( z ) = \sqrt { 1 - z ^ { 2 } }$

Find the formula for the function. -Express the area of a square as a function of its side length x.

Find the domain and range of the function. - $g ( z ) = \frac { - 5 } { \sqrt { z + 1 } }$

Find the domain and range of the function. - $F ( t ) = \frac { 6 } { \sqrt { t } }$

Quiz 1: Functions

Determine whether or not the graph is a graph of a function of x. -

Find the formula for the function. -Express the area of a circle as a function of its radius r.

Find the domain and range of the function. - f(x) =4−x2f ( x ) = 4 - x ^ { 2 }f(x) =4−x2

Determine whether or not the graph is a graph of a function of x. -

Find the formula for the function. -Express the perimeter of a square as a function of the square's side length x.

Find the formula for the function. -Express the length d of a square's diagonal as a function of its side length x.

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

Find the domain and range of the function. - F(t) =t2−3F ( t ) = t ^ { 2 } - 3F(t) =t2−3

Find the domain and range of the function. - f(x) =710+xf ( x ) = \frac { 7 } { 10 + \sqrt { x } }f(x) =10+x​7​

Find the domain and range of the function. - g(z) =−2−zg ( z ) = - 2 - \sqrt { z }g(z) =−2−z​

Find the domain and range of the function. - F(t) =109tF ( t ) = \frac { 10 } { 9 \sqrt { t } }F(t) =9t​10​

Determine whether or not the graph is a graph of a function of x. -

Determine whether or not the graph is a graph of a function of x. -

Determine whether or not the graph is a graph of a function of x. -

Find the formula for the function. -Express the perimeter of an isosceles triangle with side lengths x, 5x, and 5x as a function of the side length.

Determine whether or not the graph is a graph of a function of x. -

Find the domain and range of the function. - g(z) =1−z2g ( z ) = \sqrt { 1 - z ^ { 2 } }g(z) =1−z2​

Find the formula for the function. -Express the area of a square as a function of its side length x.

Find the domain and range of the function. - g(z) =−5z+1g ( z ) = \frac { - 5 } { \sqrt { z + 1 } }g(z) =z+1​−5​

Find the domain and range of the function. - F(t) =6tF ( t ) = \frac { 6 } { \sqrt { t } }F(t) =t​6​

Find the domain and range of the function. - $f ( x ) = 4 - x ^ { 2 }$

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

Find the domain and range of the function. - $F ( t ) = t ^ { 2 } - 3$

Find the domain and range of the function. - $f ( x ) = \frac { 7 } { 10 + \sqrt { x } }$

Find the domain and range of the function. - $g ( z ) = - 2 - \sqrt { z }$

Find the domain and range of the function. - $F ( t ) = \frac { 10 } { 9 \sqrt { t } }$

Find the domain and range of the function. - $g ( z ) = \sqrt { 1 - z ^ { 2 } }$

Find the domain and range of the function. - $g ( z ) = \frac { - 5 } { \sqrt { z + 1 } }$

Find the domain and range of the function. - $F ( t ) = \frac { 6 } { \sqrt { t } }$