Math  /  Algebra

Questionf(x)=4xf(x) = -4x and g(x)=x4g(x) = \sqrt{x-4}
1 of 2: Find the formula for (f+g)(x)(f + g)(x) and simplify your answer. Then find the domain for (f+g)(x)(f + g)(x). Round your answer to two decimal places, if necessary.
(f+g)(x)=(f + g)(x) =
Domain ==

Studdy Solution

STEP 1

What is this asking? We need to add two functions, f(x)f(x) and g(x)g(x), together and then figure out which xx values we can plug into the combined function. Watch out! Remember that the domain of the combined function is limited by the domains of the original functions!

STEP 2

1. Combine the functions
2. Find the domain of the combined function

STEP 3

We're given f(x)=4xf(x) = -4x and g(x)=x4g(x) = \sqrt{x-4}.

STEP 4

Adding the functions means (f+g)(x)=f(x)+g(x)(f+g)(x) = f(x) + g(x).
Let's **substitute** the expressions for f(x)f(x) and g(x)g(x) to get (f+g)(x)=4x+x4(f+g)(x) = -4x + \sqrt{x-4}.
Boom!

STEP 5

The function f(x)=4xf(x) = -4x is a simple line.
We can plug *any* real number in for xx, so its domain is all real numbers.

STEP 6

Now, g(x)=x4g(x) = \sqrt{x-4} is a little trickier.
We can't take the square root of a negative number, so we need x40x - 4 \ge 0. **Add** 44 to both sides of the inequality to get x4x \ge 4.
So, the domain of g(x)g(x) is all xx values greater than or equal to **4**.

STEP 7

The domain of (f+g)(x)(f+g)(x) is the intersection of the domains of f(x)f(x) and g(x)g(x).
Since f(x)f(x) is defined everywhere, we only need to consider the domain of g(x)g(x).
Therefore, the domain of (f+g)(x)(f+g)(x) is x4x \ge 4.
We write this in interval notation as [4,)[4, \infty).

STEP 8

(f+g)(x)=4x+x4(f+g)(x) = -4x + \sqrt{x-4}
Domain =[4,)= [4, \infty)

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