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

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.

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.

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.

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.

Transcript

00:01 This question gives it the function, and we're supposed to compute this function at several values.

00:06 So our function is f of x is equal to 3x squared minus x plus 2.

00:13 The first thing we want to find is f of 2.

00:16 For that, we just plug in 2 everywhere we see in x.

00:19 So we get 3 times 2 squared minus 2 plus 2.

00:23 The minus 2 plus 2 adds to 0, which cancels out.

00:26 So then we have 3 times 2 squared, which is 4.

00:30 Since 2 times 2 is 4, and then 3 times 4 is 12.

00:33 So f is 2 is equal to 12.

00:38 The next value is f of a.

00:40 A is just supposed to represent any number.

00:43 So in this case, we plug in a wherever we see x.

00:46 So that would be 3a squared minus a plus 2.

00:50 And that is the simplest we can get for our formula for f of a.

00:55 Next value is f of negative 2, same as 2, we just plug in negative 2 everywhere.

01:00 So we have 3 times negative 2 squared minus negative 2 plus 2.

01:07 Negative 2 squared is also 4.

01:10 And then minus negative 2 is plus 2.

01:13 And then we have another plus 2.

01:15 3 times 4 is 12.

01:16 And then 2 plus 2 is 4.

01:18 So we get 12 plus 4, which is equal to 16.

01:22 So f of negative 2 is 16.

01:26 The next value is f of negative a, which will work like a.

01:31 So we have three times negative a squared minus negative a plus two.

01:39 So no matter if a is a positive number or if a is a negative number, a squared or negative a squared will both be a squared.

01:50 It's going to be the same value.

01:52 Minus negative a is plus a and then plus two.

01:56 And that is f of negative a, three a squared plus a plus two.

02:03 Next value we have to plug in is a plus 1.

02:06 So for f of a plus 1, we have 3 times the quantity of a plus 1 squared minus the quantity of a plus 1 and then plus 2.

02:16 So for this one, we're going to have to foil out a binomial for the a plus 1 quantity squared.

02:24 First, we just represented as a plus 1 times a plus 1.

02:28 Then i'm going to go ahead and distribute this negative to a and 1 and add two.

02:36 So i'm going to do a foil for this binomial here.

02:39 Our first two terms are a, so a times a is a squared.

02:43 Our outer is a and one, so we multiply a times one.

02:47 Our inner is the same thing, so we add another a.

02:50 Our last two terms are one, so we just have one times one, which is one, minus a, and then this negative one plus two, that would just be a positive one.

03:02 Next, i'm going to simplify in these parentheses.

03:07 Instead of a squared plus a plus a plus a plus one, we have a squared plus two a plus one.

03:12 Then we have the negative a plus one out here.

03:14 Now i'm going to go ahead and distribute this three to all three terms.

03:19 So we have three a squared plus six a since we multiply three times two plus three minus a plus one.

03:28 Then we can do a little more combining.

03:30 We have three a squared plus 5a since we have the 6a minus 1a then we have plus 3 plus 1 so that adds to 4 and this is equal to f of a plus 1 the next value it has us compute is 2 times f of a so for this we actually just have to do 2 times whatever we found f of a b up here we already calculated f of a to be 3a squared minus a plus 2 so we put this in the parentheses, 3a squared minus a plus 2.

04:12 Now we'll just distribute the 2, 6a squared minus 2a plus 4, and that is equal to 2 times f of a.

04:25 The next value we must find is f of 2a...

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