Problem 75 Let $f(x)=3 x^{2}-x$ and \(g(x... [FREE SOLUTION]

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Chapter 2: Problem 75

Let $f(x)=3 x^{2}-x$ and $g(x)=4 x-2 .$ Find the following. $$f(x+h)-f(x)$$

Short Answer

Expert verified

$f(x+h) - f(x) = 6xh + 3h^2 - h$

Step by step solution

Identify the function f(x)

Identify that the given function is $f(x) = 3x^2 - x$.

Substitute $x + h$ into f(x)

Replace $x$ with $x + h$ in the function $f(x)$ to get $f(x+h) = 3(x+h)^2 - (x+h)$.

Expand $f(x+h)$

Expand $f(x+h)$:$f(x+h) = 3(x^2 + 2xh + h^2) - (x + h)$$= 3x^2 + 6xh + 3h^2 - x - h$.

Calculate $f(x+h) - f(x)$

Compute the difference $f(x+h) - f(x)$:\[f(x+h) - f(x) = (3x^2 + 6xh + 3h^2 - x - h) - (3x^2 - x)\]Simplify the expression:\[= 3x^2 + 6xh + 3h^2 - x - h - 3x^2 + x\]\[= 6xh + 3h^2 - h\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Substitution

The concept of function substitution involves replacing the variable in a given function with another expression. In this exercise, we start with the function $f(x) = 3x^2 - x$. To find $f(x+h)$, we substitute $x+h$ in place of $x$. This process is essential in various algebraic manipulations, helping us transform and solve more complex expressions.

To apply function substitution:

Identify the original function, here $f(x) = 3x^2 - x$.
Substitute $x + h$ into the function, yielding $f(x+h) = 3(x+h)^2 - (x+h)$.

This substitution sets the stage for further calculations, such as polynomial expansions and calculating the difference of functions, as we will discuss next.

Polynomial Expansion

Polynomial expansion involves expanding a given polynomial expression into a simplified form. After substituting $x + h$ into $f(x)$, we get $f(x+h) = 3(x+h)^2 - (x+h)$. Expanding $(x+h)^2$ allows us to simplify this expression:

First, recognise that $ (x+h)^2 $ expands to $ x^2 + 2xh + h^2 $.
Thus, $ 3(x+h)^2 $ becomes $ 3(x^2 + 2xh + h^2) = 3x^2 + 6xh + 3h^2 $.
After expansion, don't forget to also expand $ -(x+h) $, which yields $ -x - h $.

Putting all parts together, we have:
\[ f(x+h) = 3x^2 + 6xh + 3h^2 - x - h \].
Through polynomial expansion, we've simplified the substituted function, making it easier to perform further operations such as finding the difference of functions.

Difference of Functions

Finding the difference between functions means subtracting one function from another. Here, we need to compute $f(x+h) - f(x)$. With our previous expansions, this becomes more straightforward:

First, write out $ f(x+h) $ and subtract $ f(x) $ from it: \[ f(x+h) - f(x) = (3x^2 + 6xh + 3h^2 - x - h) - (3x^2 - x) \].
Next, distribute the negative sign across $ f(x) $: \[ = 3x^2 + 6xh + 3h^2 - x - h - 3x^2 + x \].
Now, cancel out the common terms ($3x^2$ and $-x$): \[ = 6xh + 3h^2 - h \].

This expression, $6xh + 3h^2 - h$, is the difference of $f(x + h)$ and $f(x)$. The process highlights the importance of substitutions and expansions in simplifying and solving algebraic expressions.

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91影视

Let \(f(x)=3 x^{2}-x\) and \(g(x)=4 x-2 .\) Find the following. $$f(x+h)-f(x)$$

Short Answer

Step by step solution

Identify the function f(x)

Substitute \(x + h\) into f(x)

Expand \(f(x+h)\)

Calculate \(f(x+h) - f(x)\)

Key Concepts

Function Substitution

Polynomial Expansion

Difference of Functions

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