Question 1

Evaluate a Function
Evaluate the function at the given value of the independent variable and simplify.
-$$h ( x ) = | x - 11 | ; \quad h ( 15 )$$

Accepted Answer

To evaluate $h(15)$, substitute $x = 15$ into the function: $h(15) = |15 - 11| = |4| = 4$.

Question 2

Evaluate a Function
Evaluate the function at the given value of the independent variable and simplify.
-$$f ( x ) = \sqrt { x + 19 } ; \quad f ( - 3 )$$

Accepted Answer

To evaluate $f(-3)$, substitute $-3$ for $x$ in the function: $f(-3) = \sqrt{-3 + 19} = \sqrt{16} = 4$.

Question 3

Determine whether the equation defines y as a function of x.
-$$x ^ { 2 } + y ^ { 2 } = 36$$

Accepted Answer

This equation defines a circle with radius 6 centered at the origin. For each value of x on the circle, there are two possible values of y, one above the x-axis and one below. Therefore, y is not a function of x.

Question 4

Determine whether the equation defines y as a function of x.
-$$y = \sqrt { 6 x - 2 }$$

Accepted Answer

To check whether the equation defines y as a function of x or not, we need to verify if for each x value there exists only one y value. In other words, for each input x, we should get only one output y. 
In this case, we can see that for any value of x, there is only one corresponding value of y given by the square root of 6x-2. Therefore, the equation defines y as a function of x.

Question 5

Determine whether the equation defines y as a function of x.
-$$y = x ^ { 3 }$$

Accepted Answer

For every value of x, there is only one corresponding value of y. Therefore, this equation defines y as a function of x.

Question 6

Evaluate a Function
Evaluate the function at the given value of the independent variable and simplify.
-$f ( x ) = 3 x ^ { 2 } - 4 x + 2 ; \quad f ( x - 1 )$

Accepted Answer

To evaluate $f(x-1)$, substitute $x-1$ for $x$ in the function $f(x) = 3x^2 - 4x + 2$. This gives $f(x-1) = 3(x-1)^2 - 4(x-1) + 2$. Expanding and simplifying, $3(x^2 - 2x + 1) - 4x + 4 + 2$ results in $3x^2 - 6x + 3 - 4x + 6 = 3x^2 - 10x + 9$.

Question 7

Determine whether the equation defines y as a function of x.
-$$x + y ^ { 3 } = 1$$

Accepted Answer

The equation defines y as a function of x since for every unique value of x, there is only one corresponding value of y that satisfies the equation. We can solve for y as follows: $$y ^ { 3 } = 1 - x $$ $$y = \sqrt [ 3 ] { 1 - x }$$ Therefore, y can be expressed solely in terms of x, indicating that y is a function of x.

Question 8

Evaluate a Function
Evaluate the function at the given value of the independent variable and simplify.
-$$f ( x ) = \frac { x ^ { 3 } + 8 } { x ^ { 2 } - 4 } ; \quad f ( 5 )$$

Accepted Answer

The answer of Evaluate a Function
Evaluate the function at the...

Question 9

Determine whether the equation defines y as a function of x.
-$$| x | - y = 6$$

Accepted Answer

This equation defines y as a function of x because every x-value has exactly one corresponding y-value. To find the corresponding y-value for a given x-value, we simply need to solve for y by subtracting |x| from both sides of the equation. For example, if x = 3, then |3| - y = 6, which simplifies to y = -3. If x = -3, then |-3| - y = 6, which also simplifies to y = -3. Therefore, there is a unique y-value for every x-value, which makes y a function of x.

Question 10

Evaluate a Function
Evaluate the function at the given value of the independent variable and simplify.
-$$f ( x ) = \frac { x ^ { 2 } - 3 } { x ^ { 3 } - 6 x } ; f ( 2 )$$

Accepted Answer

To evaluate $f(2)$, substitute $x = 2$ into the function: $f(2) = \frac{2^2 - 3}{2^3 - 6 \cdot 2} = \frac{4 - 3}{8 - 12} = \frac{1}{-4} = -\frac{1}{4}$.

Question 11

Evaluate a Function
Evaluate the function at the given value of the independent variable and simplify.
-$$f ( x ) = x ^ { 2 } - 4 ; \quad f ( x + 3 )$$

Accepted Answer

The answer of Evaluate a Function
Evaluate the function at the...

Question 12

Determine whether the equation defines y as a function of x.
-$$x = y ^ { 2 }$$

Accepted Answer

For a given value of $x$, there can be two values of $y$ (one positive and one negative) since $y^2 = x$. This violates the definition of a function, which requires each input to have exactly one output.

Question 13

Determine whether the equation defines y as a function of x.
-$$x y + 6 y = 1$$

Accepted Answer

To determine if y is a function of x, we need to check if there is only one unique y-value for each x-value. Solving for y, we get: 
$$y = \frac{1}{x+6}$$
Since there is only one possible y-value for each x-value, y is a function of x.

Question 14

Evaluate a Function
Evaluate the function at the given value of the independent variable and simplify.
-$$g ( x ) = 3 x + 1 ; \quad g ( x + 1 )$$

Accepted Answer

To evaluate the function at x + 1, we simply substitute x + 1 in place of x in the function:
g(x + 1) = 3(x + 1) + 1
= 3x + 3 + 1
= 3x + 4
Therefore, the answer is A, 3x + 4.

Question 15

Solve the problem.
-The total cost in dollars for a certain company to produce x empty jars to be used by a jelly producer is given by the function C(x)= 0.3x + 27,000. Find C(90,000), the cost of producing 90,000 jars.

Accepted Answer

To find C(90,000), substitute x = 90,000 into the function: C(90,000) = 0.3(90,000) + 27,000 = 27,000 + 27,000 = $54,000.

Question 16

Determine whether the equation defines y as a function of x.
-$$x + y ^ { 2 } = 1$$

Accepted Answer

For a given value of $x$, there can be two different values of $y$ that satisfy the equation $x + y^2 = 1$, because $y^2$ can have both a positive and a negative square root. This violates the definition of a function, which requires that each input is associated with exactly one output.

Question 17

Determine whether the equation defines y as a function of x.
-$$y = - \sqrt { x - 8 }$$

Accepted Answer

To determine if a equation defines y as a function of x, we need to check if each x value only has one corresponding y value. In the given equation, there is only one output or one answer for each possible input of x. Hence, y is a function of x in this equation.

Question 18

Solve the problem.
-The function P(x)= 0.65x - 57 models the relationship between the number of pretzels x that a certain vendor sells and the profit the vendor makes. Find P(1000), the profit the vendor makes from selling 1000
Pretzels.

Accepted Answer

To find P(1000), substitute x = 1000 into the equation: P(1000) = 0.65(1000) - 57 = 650 - 57 = $593.

Question 19

Determine whether the equation defines y as a function of x.
-$$y ^ { 2 } = 6 x$$

Accepted Answer

For a given value of $x$, the equation $y^2 = 6x$ can produce two values of $y$ (one positive and one negative), meaning a single input $x$ can result in more than one output $y$. This violates the definition of a function, which requires each input to have exactly one output.

Question 20

Evaluate a Function
Evaluate the function at the given value of the independent variable and simplify.
-$f ( x ) = 2 x + 4 ; \quad f ( 6 )$

Accepted Answer

To evaluate $f(6)$, substitute $x = 6$ into the function: $f(6) = 2(6) + 4 = 12 + 4 = 16$.

Quiz 2: Functions and Graphs

Evaluate a Function Evaluate the function at the given value of the independent variable and simplify. - $h ( x ) = | x - 11 | ; \quad h ( 15 )$

Evaluate a Function Evaluate the function at the given value of the independent variable and simplify. - $f ( x ) = \sqrt { x + 19 } ; \quad f ( - 3 )$

Determine whether the equation defines y as a function of x. - $x ^ { 2 } + y ^ { 2 } = 36$

Determine whether the equation defines y as a function of x. - $y = \sqrt { 6 x - 2 }$

Determine whether the equation defines y as a function of x. - $y = x ^ { 3 }$

Evaluate a Function Evaluate the function at the given value of the independent variable and simplify. - $f ( x ) = 3 x ^ { 2 } - 4 x + 2 ; \quad f ( x - 1 )$

Determine whether the equation defines y as a function of x. - $x + y ^ { 3 } = 1$

Evaluate a Function Evaluate the function at the given value of the independent variable and simplify. - $f ( x ) = \frac { x ^ { 3 } + 8 } { x ^ { 2 } - 4 } ; \quad f ( 5 )$

Determine whether the equation defines y as a function of x. - $| x | - y = 6$

Evaluate a Function Evaluate the function at the given value of the independent variable and simplify. - $f ( x ) = \frac { x ^ { 2 } - 3 } { x ^ { 3 } - 6 x } ; f ( 2 )$

Evaluate a Function Evaluate the function at the given value of the independent variable and simplify. - $f ( x ) = x ^ { 2 } - 4 ; \quad f ( x + 3 )$

Determine whether the equation defines y as a function of x. - $x = y ^ { 2 }$

Determine whether the equation defines y as a function of x. - $x y + 6 y = 1$

Evaluate a Function Evaluate the function at the given value of the independent variable and simplify. - $g ( x ) = 3 x + 1 ; \quad g ( x + 1 )$

Solve the problem. -The total cost in dollars for a certain company to produce x empty jars to be used by a jelly producer is given by the function C(x) = 0.3x + 27,000. Find C(90,000) , the cost of producing 90,000 jars.

Determine whether the equation defines y as a function of x. - $x + y ^ { 2 } = 1$

Determine whether the equation defines y as a function of x. - $y = - \sqrt { x - 8 }$

Solve the problem. -The function P(x) = 0.65x - 57 models the relationship between the number of pretzels x that a certain vendor sells and the profit the vendor makes. Find P(1000) , the profit the vendor makes from selling 1000 Pretzels.

Determine whether the equation defines y as a function of x. - $y ^ { 2 } = 6 x$

Evaluate a Function Evaluate the function at the given value of the independent variable and simplify. - $f ( x ) = 2 x + 4 ; \quad f ( 6 )$

Quiz 2: Functions and Graphs

Evaluate a Function Evaluate the function at the given value of the independent variable and simplify. - h(x) =∣x−11∣;h(15) h ( x ) = | x - 11 | ; \quad h ( 15 ) h(x) =∣x−11∣;h(15)

Evaluate a Function Evaluate the function at the given value of the independent variable and simplify. - f(x) =x+19;f(−3) f ( x ) = \sqrt { x + 19 } ; \quad f ( - 3 ) f(x) =x+19​;f(−3)

Determine whether the equation defines y as a function of x. - x2+y2=36x ^ { 2 } + y ^ { 2 } = 36x2+y2=36

Determine whether the equation defines y as a function of x. - y=6x−2y = \sqrt { 6 x - 2 }y=6x−2​

Determine whether the equation defines y as a function of x. - y=x3y = x ^ { 3 }y=x3

Evaluate a Function Evaluate the function at the given value of the independent variable and simplify. - f(x) =3x2−4x+2;f(x−1) f ( x ) = 3 x ^ { 2 } - 4 x + 2 ; \quad f ( x - 1 ) f(x) =3x2−4x+2;f(x−1)

Determine whether the equation defines y as a function of x. - x+y3=1x + y ^ { 3 } = 1x+y3=1

Evaluate a Function Evaluate the function at the given value of the independent variable and simplify. - f(x) =x3+8x2−4;f(5) f ( x ) = \frac { x ^ { 3 } + 8 } { x ^ { 2 } - 4 } ; \quad f ( 5 ) f(x) =x2−4x3+8​;f(5)

Determine whether the equation defines y as a function of x. - ∣x∣−y=6| x | - y = 6∣x∣−y=6

Evaluate a Function Evaluate the function at the given value of the independent variable and simplify. - f(x) =x2−3x3−6x;f(2) f ( x ) = \frac { x ^ { 2 } - 3 } { x ^ { 3 } - 6 x } ; f ( 2 ) f(x) =x3−6xx2−3​;f(2)

Evaluate a Function Evaluate the function at the given value of the independent variable and simplify. - f(x) =x2−4;f(x+3) f ( x ) = x ^ { 2 } - 4 ; \quad f ( x + 3 ) f(x) =x2−4;f(x+3)

Determine whether the equation defines y as a function of x. - x=y2x = y ^ { 2 }x=y2

Determine whether the equation defines y as a function of x. - xy+6y=1x y + 6 y = 1xy+6y=1

Evaluate a Function Evaluate the function at the given value of the independent variable and simplify. - g(x) =3x+1;g(x+1) g ( x ) = 3 x + 1 ; \quad g ( x + 1 ) g(x) =3x+1;g(x+1)