Question

The domain of the function:

$f(x) = \sqrt {log_{10}(\frac{5x-x²}{4}) }$
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Answer 1

Solution!!

$f(x) = \sqrt {\log_{10}\left(\dfrac{5x-x²}{4}\right) }$

Separate the function into parts to determine the domain of each part.

$\to \sqrt {\log_{10}\left(\dfrac{5x-x²}{4}\right) }$

$\to \log_{10}\left(\dfrac{5x-x²}{4}\right)$

$\to 5x-x²$

Let's solve one by one.

$\to \sqrt {log_{10}\left(\dfrac{5x-x²}{4}\right) }$

The domain of an even root function are all values of x for which the radicand is positive or 0.

$\sqrt {\log_{10}\left(\dfrac{5x-x²}{4}\right) }=0$

$\log_{10}\left(\dfrac{5x-x²}{4}\right) =0$

$\dfrac{5x-x²}{4}=1$

$5x-x²=4$

$5x-x²-4=0$

$-x²+5x-4=0$

$x²-5x+4=0$

$x²-x-4x+4=0$

$x(x-1)-4(x-1)=0$

$(x-1)(x-4)=0$

$x=1\:and\:x=4$

$x\in [1,4]$

$\to \log_{10}\left(\dfrac{5x-x²}{4}\right)$

The domain of a logarithmic function are all values of x for which the argument is positive.

$x\in [0,5]$

$\to 5x-x²$

The domain of a quadratic function is the set of all real numbers.

$x\in R$

Now, we have to find the intersection.

$x\in [1,4]$

Answer 2

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Step-by-step explanation:

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