QuestionGiven the functions:
$f(x) = (x-2)^2$
$g(x) = x+1$
Find the expression for $(f-g)(x)$ .

Studdy Solution

STEP 1

What is this asking? We need to find a new function that's the difference between two given functions! Watch out! Don't forget to distribute the negative sign correctly when subtracting $g(x)$ !

STEP 2

1. Define the functions
2. Set up the difference
3. Expand and simplify

STEP 3

Alright, so we're given two functions, $f(x)$ and $g(x)$ .
Let's write them down clearly so we know what we're working with.
We have $f(x) = (x-2)^2$ and $g(x) = x+1$ .
Super important to keep these in mind!

STEP 4

Now, we want to find $(f-g)(x)$ .
This just means we want to find a new function that represents the difference between $f(x)$ and $g(x)$ .
So, we can write $(f-g)(x) = f(x) - g(x)$ .
Makes sense, right?
We're just subtracting the functions!

STEP 5

Let's plug in what we know about $f(x)$ and $g(x)$ into our expression.
We get $(f-g)(x) = (x-2)^2 - (x+1)$ .
Notice those parentheses around $x+1$ . Super important because we're subtracting the whole function $g(x)$ !

STEP 6

First, let's expand $(x-2)^2$ .
Remember, this means $(x-2) \cdot (x-2)$ .
Using the FOIL method (First, Outer, Inner, Last), we get $x \cdot x - 2 \cdot x - 2 \cdot x + (-2) \cdot (-2)$ , which simplifies to $x^2 - 4x + 4$ .

STEP 7

Now, let's substitute this back into our expression for $(f-g)(x)$ : $(f-g)(x) = x^2 - 4x + 4 - (x+1)$ .

STEP 8

Time to distribute that negative sign!
Remember, subtracting $(x+1)$ is the same as adding $(-1) \cdot (x+1)$ .
So, we have $x^2 - 4x + 4 + (-1) \cdot x + (-1) \cdot 1$ , which simplifies to $x^2 - 4x + 4 - x - 1$ .

STEP 9

Finally, let's combine like terms.
We have $x^2$ , $-4x$ and $-x$ , and then $4$ and $-1$ .
Combining these gives us our final simplified expression: $x^2 - 5x + 3$ !

STEP 10

$(f-g)(x) = x^2 - 5x + 3$

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