If f(x)=3 x^2-x+2, find f(2), f(-2), f(a), f(-a), f(a+1), 2 f(a), f(2 a), f(a^2),[f(a)]^2, and f

If f(x)=3 x^2-x+2, find f(2), f(-2), f(a), f(-a), f(a+1), 2...

Key Concepts

Polynomial Function

A polynomial function is an expression consisting of variables and coefficients that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Understanding the structure of a polynomial, such as recognizing the degree and the coefficients, is crucial in analyzing and evaluating the function.

Function Evaluation

Function evaluation involves substituting a specific value into the function to compute the output. This process is fundamental in understanding how the function behaves for different inputs, whether the inputs are numbers, algebraic expressions, or other functions.

Substitution

Substitution is the process of replacing a variable within an expression or function with another expression or number. This technique is used to simplify expressions, evaluate the function at particular points (including expressions like a+1, 2a, a+h), and to explore the function's behavior under various changes to its input.

Algebraic Simplification

Algebraic simplification entails manipulating expressions to combine like terms, clear parentheses, and rewrite expressions in a more manageable form. This is essential after substituting specific values into functions to achieve the final simplified result.

Function Composition and Transformation

Function composition and transformation involve modifying the input of a function by applying operations such as scaling, shifting, or squaring. Evaluating the function at expressions like a^2 or a+h illustrates how changes to the input affect the output of a function, highlighting the concept of function transformations.

Transcript

00:01 Okay, this question is about how we use function notation and mathematics.

00:06 We have a function, in this case, it's called f of x.

00:10 X is the domain, the values that we plug in here, the inputs, x are the values of the domain, and then we have this function that says 3x squared minus x plus 2.

00:22 So the way we read that, whatever we have here, f of x equals 3x squared, squared minus x plus two.

00:33 Whatever you have for x, you're going to square it.

00:37 Take that square, multiply it by three.

00:39 You're then going to subtract the x value, and you're going to add two.

00:43 So for the first problem, it says find f of two.

00:47 I always tell students, whenever you substitute a value in for x into a function, use parentheses.

00:54 If you don't need the parentheses, then they'll simplify out when you're doing the problem.

01:00 But if you need parentheses and you don't have them, you could get the problem wrong.

01:05 So take two and square.

01:06 You get four, three times four is 12.

01:10 Subtract two and add two.

01:12 Well, those would cancel out.

01:14 And you just get that f of 2 is equal to 12.

01:17 And what we've done is we've evaluated the function f of x, and we found what f of 2 is.

01:24 We like that notation a lot.

01:26 F of 2 equals 12.

01:28 What that tells you is the point 2 comma 12 is on f of x.

01:34 All right.

01:35 So let's try the next part of the question.

01:40 It said for f of x equals 3x squared minus x minus or sorry plus two.

01:48 Find f of negative two.

01:51 Same thing.

01:53 Plug in negative 2 for x in a parenthesis.

01:57 Notice i'm just replacing.

01:58 That function, or sorry, in that function f of x, i'm just replacing x with the parentheses.

02:06 Sorry, negative 2 there.

02:07 Negative 2 squared is 4, 3 times 4 is 12, but now we have minus a negative 2.

02:13 That becomes plus 2 plus 2, and now we get that f of negative 2 is 16.

02:19 Again, a nice notation, f of negative 2 is 16.

02:23 What that communicates is that the point negative 2 comma 16 is on the graph.

02:31 All right, or sorry, is on the function.

02:34 So the other ones i've got to go back and look at because i won't remember all of them.

02:41 So let me just see, f of a, f of negative a.

02:43 All right.

02:45 Let's try that one.

02:47 This is where a lot of students will get hung up.

02:50 What is f of a? right.

02:52 Well, so let's look at the function again.

02:54 F of x is equal to 3x squared minus x plus two.

03:00 F of a literally means replace wherever you see x with an a and get 3a squared minus a plus two.

03:11 That's it.

03:12 And some students will get to that point and they'll be like, that's all i have to do.

03:17 Well, what's a? and it's like, but we don't know what a is.

03:19 Doesn't matter what a is.

03:21 A is a.

03:21 So whatever you want to plug in, doesn't matter if you can't simplify it, you're done.

03:28 If we take a look at f of negative a.

03:31 Now, i didn't use parentheses there.

03:33 I got a little sloppy.

03:34 I will use parentheses here.

03:36 So three, replacing x with negative a, replacing x with negative a.

03:44 Now when i simplify, it doesn't matter what a is.

03:47 If it's negative a and i square it, i will get positive a squared.

03:52 So this is 3a squared, minus a negative a becomes plus a and then plus two.

03:59 So notice that f of a and f of negative a are different expressions.

04:05 And that's as far as that one can be simplified.

04:09 F of a plus one.

04:11 So f of a plus one.

04:14 Again, we've got this function.

04:16 F of x equals 3x squared minus x plus two.

04:23 If i take f of a plus 1, you know, plug in a plus 1 wherever i see x.

04:36 I'm going to use parentheses to make sure i don't make any mistakes.

04:40 And now i just do some algebra.

04:42 When you square a plus 1, many of you may have memorized this formula for squaring a binomial.

04:49 Don't forget, when you square a binomial, you're taking a plus b times a plus b...

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