9 Functions that you Need to Know.
Take a deep breath and visualize the math...

Let's get started.
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What does the function of f(x)=x look like?

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What does the function of f(x)=x² look like?

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What does the function of f(x)=x³ look like?

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What does the function f(x)=√(x) look like? That means square root.

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Practice: If f(x)=x³, and our new function g(x)=f(x)... what kind of transformation is happening?

Reflection.

Practice: If f(x)=x³, and our new function g(x)=f(x)... This is reflection. What rule will this be? (x,y)>( , )

Now, f(x) is f(x)... which means all the y is now negative. (Because f(x) really means y). So... the new rule is (x,y)>(x,y)... this means reflection about the xaxis. You can visualize this if all y's are now y's, that means tops became on bottom and vice versa.

What does y= x look like?

Like a V shape, all is HAPPY in an absolute world.

What does the function f(x)=(2x+3)/(x+1) look like? (Or any other rational expression for that matter)... What is a rational expression again?

A rational expression is a fraction where the numerator &/or the demon. is a polynomial. You will get monomials and/or binomials maximum). Remember NPVs.

What does f(x) = 1/x look like?

What is the NPV? X cannot be... what does that create, and impact the look of this function?

What does f(x)=2^x look like?

What word should be ringing in your ears when you see a function like this? A whole unit dedicated to this type of function! How do we make it workable? This is an example of an "EXPONENTIAL" function. x=log_2(y)

Practice: How do we make f(x)=5^x more workable? Why isn't it currently workable?

We hate letters in the exponent, we have no tricks up our sleeves to make this work! So... we use logs. y=5^x, is now... x=log_5(y). X and Y swap places, and 5 is still 'base' in either version.

Practice: What is the relationship between logs and exponentials?

logs are the inverse of exponentials. This means x becomes y and y becomes x. ORRRRR this means, reflection across the line y=x.

What is general name of this function, and shape? y=3^x?

The name is an "exponential function". It looks like a J. Whatever you think of, Increasing exponentially. Note that the x is up in the exponent. We don't like functions like this usually. We can convert to logs to make more workable, or use logs to find the inverse of this kind of function. Do you remember how to convert to either type?

What is the general name of this function, and shape? y=log_2(x)

This is a "logarithmic" function. Looks like x^2 but, starts lower. Logs have a whole set of special rules to work with them. Like speaking a different dialect of math. They are VERY related to exponentials. Logs can be seen in scales that relate to pH, decibels, richter scale, fstop (photography), information theory.
