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

step 1 Consider f of x equal to 3x squared minus x plus 2. We want to find f at x equal to 2, f at x eq

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

Key Concepts

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.

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.

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.

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.

Transcript

00:01 Consider f of x equal to 3x squared minus x plus 2.

00:08 We want to find f at x equal to 2, f at x equal to minus 2, f at a, f at minus a, f evaluated at a plus 1, 2 times f of a, f of 2a, f of a squared, f of a squared, and lastly f of a plus h.

00:56 Let's start with f of 2.

00:58 We evaluate this by replacing x with 2 in our expression, and so we obtain 3 times 4 minus 2 plus 2, simplifying to 12.

01:11 F of minus 2 will also give us 3 times 4 minus 2, plus 2 minus 2, excuse me, plus 2 plus 2, which simplifies to 20.

01:28 F of a.

01:32 Replacing.

01:33 So we'll see x with a in our expression will simply give us 3a squared minus a plus 2.

01:40 F of minus a will give us 3a squared plus a plus 2, so we notice that the minus signs within the power of 2 we can't allow, but not within the minus a term.

01:53 Because it's an even function, it's an odd function.

02:01 F of a plus 1 develops us 3 times a plus 1 squared minus a.

02:09 Plus 2.

02:09 Now let's develop this, and we obtain 3a squared plus 6a plus 9 minus a plus 2, which becomes 3a squared plus 5a plus 11.

02:40 Then there's 2 times f of a.

02:43 So let's take f of a here, multiply it by 2, and we obtain term by term.

02:48 We obtain 6a squared.

02:50 So we get f of a squared minus 2a plus 4.

02:54 F of 2a, that's a little bit different...