# Algebra 1 : Functions and Lines

## Example Questions

### Example Question #6 : How To Find Inverse Variation

varies directly with , and inversely with the square root of .

If  and , then .

Find  if  and .

Explanation:

The variation equation can be written as below. Direct variation will put in the numerator, while inverse variation will put in the denominator. is the constant that defines the variation.

To find constant of variation, , substitute the values from the first scenario given in the question.

We can plug this value into our variation equation.

Now we can solve for given the values in the second scenario of the question.

### Example Question #1 : Indirect Proportionality

varies inversely as the square root of . If , then . Find  if  (nearest tenth, if applicable).

Explanation:

The variation equation is  for some constant of variation .

Substitute the numbers from the first scenario to find :

The equation is now .

If , then

### Example Question #1 : How To Find Inverse Variation

varies inversely as the square of . If , then . Find  if  (nearest tenth, if applicable).

Explanation:

The variation equation is  for some constant of variation .

Substitute the numbers from the first scenario to find :

The equation is now .

If , then

### Example Question #11 : Indirect Proportionality

The current, in amperes, that a battery provides an electrical object is inversely proportional to the resistance, in ohms, of the object.

A battery provides 1.2 amperes of current to a flashlight whose resistance is measured at 20 ohms. How much current will the same battery supply to a flashlight whose resistance is measured at 16 ohms?

Explanation:

If  is the current and  is the resistance, then we can write the variation equation for some constant of variation :

or, alternatively,

To find  , substitute :

The equation is . Now substitute  and solve for :

### Example Question #1 : How To Find Inverse Variation

If  is inversely proportional to  and knowing that  when , determine the proportionality constant.

Explanation:

The general formula for inverse proportionality for this problem is

Given that  when , we can find  by plugging them into the formula.

Solve for  by multiplying both sides by 5

So .

### Example Question #921 : Functions And Lines

The number of days needed to construct a house is inversely proportional to the number of people that help build the house. It took 28 days to build a house with 7 people. A second house is being built and it needs to be finished in 14 days. How many people are needed to make this happen?

Explanation:

The general formula of inverse proportionality for this problem is

where  is the number of days,  is the proportionality constant, and  is number of people.

Before finding the number of people needed to build the house in 14 days, we need to find . Given that the house can be built in 28 days with 7 people, we have

Multiply both sides by 7 to find .

So . Thus,

Now we can find the how many people are needed to build the house in 14 days.

Solve for . First, multiply by  on both sides:

Divide both sides by 14

So it will take 14 people to complete the house in 14 days.

### Example Question #921 : Functions And Lines

The number of days to construct a house varies inversely with the number of people constructing that house. If it takes 28 days to construct a house with 6 people helping out, how long will it take if 20 people are helping out?

Explanation:

The statement, 'The number of days to construct a house varies inversely with the number of people constructing that house' has the mathematical relationship , where D is the number of days, P is the number of people, and k is the variation constant. Given that the house can be completed in 28 days with 6 people, the k-value is calculated.

This k-value can be used to find out how many days it takes to construct a house with 20 people (P = 20).

So it will take 8.4 days to build a house with 20 people.

### Example Question #5 : Inverse Functions

Which one of the following functions represents the inverse of

A)

B)

C)

D)

E)

C)

A)

D)

B)

E)

C)

Explanation:

Given

Hence

Interchanging  with  we get:

Solving for  results in .

### Example Question #161 : Algebraic Functions

The amount of lonliness you feel varies inversely with the number of friends you have. If having 4 friends gives you a 10 on the lonliness scale, how much lonliness do you feel if you have 100 friends?

Explanation:

If lonliness and friends are inversely proportional, we can set up an equation to solve for some missing constant, k. To make things easier to write, let's use the variable L for the lonliness scale, and the variable F for how many friends you have. First we can just set up the equation:

. We know that having 4 friends gives you a 10 on the lonliness scale, so we can plug those values in to start solving for k:

to solve, multiply both sides by 4:

Now that we know the constant is 40, we can figure out how much lonliness, L, corresponds to having 100 friends by setting up a new formula. We are still generally using , and we know k is 40 and F is 100:

We can divide to get

### Example Question #162 : Algebraic Functions

The number of lights on in your room varies inversely with the number of monsters you think are under the bed. If you only have 1 light on in your room, you're pretty sure that there are 15 monsters under the bed. How many monsters do you suspect if you turn on 4 more lights [for a total of 5]?