Math  /  Algebra

Question2. Let f(x)=20xf(x)=\sqrt{20-x} and g(x)=10x4g(x)=\sqrt{10 x-4}. Let h(x)=f(x)g(x)h(x)=f(x) g(x). Find h(4)h(4).

Studdy Solution

STEP 1

What is this asking? We need to find the value of the function h(x)=f(x)g(x) h(x) = f(x) \cdot g(x) when x=4 x = 4 . Watch out! Don't forget to check if the expressions inside the square roots are non-negative for the given x x .

STEP 2

1. Evaluate f(4) f(4)
2. Evaluate g(4) g(4)
3. Calculate h(4) h(4)

STEP 3

First, let's **calculate** f(4) f(4) .
The function f(x) f(x) is defined as f(x)=20x f(x) = \sqrt{20 - x} .

STEP 4

Substitute x=4 x = 4 into the function:
f(4)=204f(4) = \sqrt{20 - 4}

STEP 5

Simplify the expression inside the square root:
f(4)=16f(4) = \sqrt{16}

STEP 6

And finally, calculate the square root:
f(4)=4f(4) = 4

STEP 7

Now, let's **calculate** g(4) g(4) .
The function g(x) g(x) is defined as g(x)=10x4 g(x) = \sqrt{10x - 4} .

STEP 8

Substitute x=4 x = 4 into the function:
g(4)=1044g(4) = \sqrt{10 \cdot 4 - 4}

STEP 9

Simplify the expression inside the square root:
g(4)=404g(4) = \sqrt{40 - 4}

STEP 10

Simplify further:
g(4)=36g(4) = \sqrt{36}

STEP 11

And calculate the square root:
g(4)=6g(4) = 6

STEP 12

Finally, let's **calculate** h(4) h(4) using the values we found for f(4) f(4) and g(4) g(4) .

STEP 13

The function h(x) h(x) is defined as h(x)=f(x)g(x) h(x) = f(x) \cdot g(x) .

STEP 14

Substitute the values we found:
h(4)=f(4)g(4)=46h(4) = f(4) \cdot g(4) = 4 \cdot 6

STEP 15

And multiply:
h(4)=24h(4) = 24

STEP 16

The value of h(4) h(4) is 24\mathbf{24}.

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