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Consider the function represented by the equation 6c = 2p – 10. Write the equation in function notation, where c is the independent variable.

f(c) = 3c + 5
f(p) = one-thirdp + five-thirds
f(p) = 3c + 5

7 answers

To write the equation \( 6c = 2p - 10 \) in function notation with \( c \) as the independent variable, we need to solve the equation for \( p \).

Starting with the equation:

\[ 6c = 2p - 10 \]

First, isolate \( p \) by adding 10 to both sides:

\[ 6c + 10 = 2p \]

Next, divide both sides by 2:

\[ p = 3c + 5 \]

Now we can write this in function notation:

\[ f(c) = 3c + 5 \]

So the correct function notation for this equation, with \( c \) as the independent variable, is:

\[ f(c) = 3c + 5 \]

f(p) = 3c + 5 YAY

Marlene rides her bike at a rate of 16 miles per hour. The time in hours that she rides is represented by the variable t, and the distance she rides is represented by the variable d. This relationship is modeled with distance d as a function of time t.

Which statements are true of the scenario? Select two answers..

The independent variable, the input, is the variable d, representing distance.
The distance traveled depends on the amount of time Marlene rides her bike.
The initial value of the scenario is 16 miles per hour.
The equation t = d + 16 represents the scenario.
The function f(t) = 16t represents the scenario.

Let's analyze the scenario involving Marlene riding her bike at a rate of 16 miles per hour, where the time \( t \) and distance \( d \) are related.

  1. The independent variable, the input, is the variable d, representing distance.

    • This statement is false. In this scenario, the time \( t \) is the independent variable because distance depends on the time she rides.

  2. The distance traveled depends on the amount of time Marlene rides her bike.

    • This statement is true. The distance \( d \) is a function of time \( t \); as time increases, the distance traveled also increases.

  3. The initial value of the scenario is 16 miles per hour.

    • This statement is true. The constant speed (rate) at which Marlene rides her bike, 16 miles per hour, is the initial value or rate of distance traveled per unit of time.

  4. The equation t = d + 16 represents the scenario.

    • This statement is false. The correct relationship derived from speed, distance, and time is \( d = 16t \), not \( t = d + 16 \).

  5. The function f(t) = 16t represents the scenario.

    • This statement is true. The function \( f(t) = 16t \) describes the distance \( d \) as a function of time \( t \) with the speed of 16 miles per hour.

Based on the analysis, the two true statements are:

  • The distance traveled depends on the amount of time Marlene rides her bike.
  • The function f(t) = 16t represents the scenario.
1. The function V(r) = four-thirds pi r cubed can be used to find the volume of air inside a basketball given its radius. What does V(r) represent?

the radius of the basketball when the volume is V
the volume of the basketball when the radius is r
the volume of the basketball when the radius is V
the radius of the basketball when the volume is r

2. The function f(x) is given by the set of ordered pairs.

{(1,0), (–10,2), (0,6), (3,17), (–2,–1)}

Which equation is true?

f of negative 10 = 1
f of 2 = negative 10
f of 0 = 6
f of 1 = negative 10

3. Consider the function represented by the table.

The ordered pair given in the bottom row can be written using function notation as

f(9)=5.
f(5)=9.
f(5, 9)=14.
f(9, 5)=14.

5. On a coordinate plane, a straight line with a negative slope, labeled f of x, crosses the y-axis at (0, 4), and the x-axis at (4, 0).
Which is true regarding the graphed function f(x)?

f of 0 = 3
f of 5 = negative 1
f of 3 = 2
f of 2 = negative 2

5. Consider the function represented by the equation y minus 6 x minus 9 = 0. Which answer shows the equation written in function notation with x as the independent variable?

f of x = 6 x + 9
f of x = one-sixth x + three-halves
f of y = 6 y + 9
f of y = one-sixth y + three-halves

6. The table represents a function.

A 2-column table with 4 rows. The first column is labeled x with entries negative 4, negative 1, 3, 5. The second column is labeled f of x with entries negative 2, 5, 4, negative 8.
What is f(5)?

–8
–1
1
8

7. The table represents a function.

A 2-column table with 4 rows. The first column is labeled x with entries negative 6, negative 2, 0, 3. The second column is labeled f of x with entries 3, 1, 4, negative 2.
What is f(–2)?

–3
–1
1
3

8. The point (–3, –5) is on the graph of a function. Which equation must be true regarding the function?

f(–3) = –5
f(–3, –5) = –8
f(–5) = –3
f(–5, –3) = –2

9. The height of a rocket a given number of seconds after it is released is modeled by h (t) = negative 16 t squared + 32 t + 10. What does t represent?

the number of seconds after the rocket is released
the initial height of the rocket
the initial velocity of the rocket
the height of the rocket after t seconds

10. If point (4, 5) is on the graph of a function, which equation must be true?

f(5)=4
f(5,4)=9
f(4)=5
f(5,4)=1

Let’s go through each question one by one:

1. What does \( V(r) \) represent?

Answer: The volume of the basketball when the radius is \( r \).

2. Which equation is true for the function given by the set of ordered pairs?

Ordered pairs: {(1,0), (–10,2), (0,6), (3,17), (–2,–1)} True statement: \( f(0) = 6 \)

3. The ordered pair given in the bottom row can be written using function notation as:

Answer: \( f(9) = 5 \).

4. Which is true regarding the graphed function \( f(x) \)?

Given points: The line crosses the y-axis at (0, 4) and the x-axis at (4, 0). True statement: \( f(3) = 2 \) (Using the slope formula and substituting values will show this is accurate).

5. Rewrite the equation \( y - 6x - 9 = 0 \) in function notation with \( x \) as the independent variable.

Answer: \( f(x) = 6x + 9 \).

6. What is \( f(5) \)?

Given table entries:

  • \( f(5) \) corresponds to the x-value of 5, which gives \( -8 \). Answer: \( -8 \).

7. What is \( f(–2) \)?

Given table entries:

  • \( f(-2) \) corresponds to the x-value of –2, which gives \( 1 \). Answer: \( 1 \).

8. Which equation must be true regarding the function if the point (–3, –5) is on the graph?

Answer: \( f(–3) = –5 \).

9. What does \( t \) represent in the rocket height model \( h(t) = -16t^2 + 32t + 10 \)?

Answer: The number of seconds after the rocket is released.

10. If point (4, 5) is on the graph of a function, which equation must be true?

Answer: \( f(4) = 5 \).

To summarize:

  1. The volume of the basketball when the radius is \( r \).
  2. \( f(0) = 6 \).
  3. \( f(9) = 5 \).
  4. \( f(3) = 2 \).
  5. \( f(x) = 6x + 9 \).
  6. \( -8 \).
  7. \( 1 \).
  8. \( f(–3) = –5 \).
  9. The number of seconds after the rocket is released.
  10. \( f(4) = 5 \).
90!!

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