Chapter 7.4-7.5: Functions and Graphs

Functions and Graphs

Chapter 7.4-7.5

The Algebra of Functions

Multiple Choice

Let

$$f\left(x\right)=x-2\ \text{and }g\left(x\right)=x-7$$  

find  $$\left(f+g\right)\left(x\right)$$  

• 2x + 9 

• 2x - 9

• x - 9

• 2x - 5

Multiple Choice

Let

$$f\left(x\right)=x-2\ \text{and }g\left(x\right)=x-7$$  

find  $$\left(f-g\right)\left(x\right)$$  

• 2x - 5

• 2x - 9

• 5

• -9

Multiple Choice

Let

$$f\left(x\right)=x-2\ \text{and }g\left(x\right)=x-7$$  

find  $$\left(f\cdot g\right)\left(5\right)$$  

• -6

• $$x^2-9x+14$$

• 1

• -1

Multiple Choice

Let

$$f\left(x\right)=x-2\ \text{and }g\left(x\right)=x-7$$  

find  $$\left(f\text{/}g\right)\left(x\right)$$  

• $$\frac{x-2}{x-7}$$  

• $$\frac{x-7}{x-2},\ x\ne2$$  

•   $$\frac{x-2\ }{x-7};\ x\ne7$$

• $$\frac{2}{7}$$  

Solving for a Given Variable

(Continued)

Multiple Choice

Solve for  $$y$$  

$$\frac{2}{k}+\frac{1}{y}=3$$  

• $$y=\frac{3ky-k}{2}$$  

• $$y=\frac{2y+k}{3k}$$  

• $$y=-\frac{k}{3k-2}$$  

• $$y=\frac{k}{3k-2}$$  

Direct Variation

• Equation of Variation: y=kx or f(x)=kx

• Variation constant: k

• y varies directly as x

• y is proportional to x 

• Whatever is on the left of the phrase "varies directly as" or "is proportional to" goes on the left of your equal sign

• Whatever is on the right the phrase "varies directly as" or "is proportional to" is on the right of your equal sign (with k)

The speed of a train of ocean waves varies directly as the swell period. Waves with a swell period of 12sec are traveling 21mph.

• the key phrase "varies directly as" tells me to use direct variation, or y=kx 

• "the speed of a train of ocean waves" is on the left "varies directly as" so it goes on the left of my equal sign

• "the swell period" comes after "varies directly as" so it goes on the right of my equal sign

• Now, we need to find our k

The speed of a train of ocean waves varies directly as the swell period. Waves with a swell period of 12sec are traveling 21mph.

The speed of a train of ocean waves varies directly as the swell period. Waves with a swell period of 12sec are traveling 21mph.  How fast are waves with a swell period of 20sec traveling?

Multiple Choice

The score on an entrance exam varies directly as the number of correct answers. If the score is 40 when 8 correct answers are given, what is the score when 10 correct answers are given?

• 12

• 2

• 5

• 50

Inverse Variation

Multiple Choice

The time it takes to stuff envelopes for a charity’s annual campaign varies inversely as the number of volunteers. If it takes 12 volunteers 4 hours to complete the job, what is the variation constant?

• $$k=\frac{1}{3}$$

• $$k=3$$

• $$k=48$$

• $$k=16$$

Multiple Choice

The time it takes to stuff envelopes for a charity’s annual campaign varies inversely as the number of volunteers. If it takes 12 volunteers 4 hours to complete the job, what is my equation of variation?

• $$f\left(x\right)=\frac{48}{x}$$  

• $$f\left(x\right)=48x$$  

• $$f\left(x\right)=3x$$  

• $$f\left(x\right)=\frac{x}{3}$$  

The time it takes to stuff envelopes for a charity’s annual campaign varies inversely as the number of volunteers. If it takes 12 volunteers 4 hours to complete the job, what is the variation constant?

We know we will use the equation y = k/x, since we see the key phrase "varies inversely as".

Since "the time it takes to stuff envelopes" comes before the phrase "varies inversely as", it is our y.

Since "the number of volunteers" comes after the phrase "varies inversely as", it is our x.

So, we have: 4 = k/12 or k = 48

Therefore, our equation of variation is y=48/x

Multiple Choice

The time it takes to stuff envelopes for a charity’s annual campaign varies inversely as the number of volunteers. If it takes 12 volunteers 4 hours to complete the job, how long will it take 5 volunteers to complete the same job?

• $$\frac{5}{3}\text{hrs}$$  

• 240hrs

• 15hrs

• 9.6hrs

The time it takes to stuff envelopes for a charity’s annual campaign varies inversely as the number of volunteers. If it takes 12 volunteers 4 hours to complete the job. How long will it take 5 volunteers to do the same job?

Using our equation of variation is y=48/x, we substitute 5 for x and solve for y.

y=48/5 or y = 9.6

This means it will take 5 volunteers 9.6 hours to do the same job it would take 12 volunteers to do.

Joint and Combined Variation

• Joint variation: y varies jointly as x and z: y=kxz 

• Combined variation uses both direct and inverse variation (see example below)

• *Remember: Whatever comes after the word "inverse" goes in the denominator

Multiple Choice

Find the equation of variation for the following:

y varies inversely as x and z and directly as the square of w, and y = 45 when x = 4, z = 5, and w = 3.

• $$y=\frac{81xz}{4w^2}$$

• $$y=\frac{100w^2}{xz}$$

• $$y=\frac{w^2xz}{4}$$