Question

f(x)=x-4/4-x find domain and range

Answer 1

Domain is the set of all possible x values .

So x-4/4-x is only possible if 4-x is not equal to 0 .

Then at x=4 f(x) is not possible .

So domain of f(x) is set of all real numbers except 4.

Now range is the set of all y values possible .

Thus simplifying f(x) we get,

x-4/(-1)(x-4)

= -1

Thus range of f(x) is -1 as that is the only possible y value no matter what x is .

Hope this helps.

Answer 2

The domain and range of the real function f defined by f(x)= $\frac{4-x}{x-4}$  is given by, Domain = $\mathbf{R}-\{4\}$, Range = $\{-1\} .$

How do you find the domain and range of a given function?

• To find the domain and range, we simply solve the equation y = f(x) to determine the values of the independent variable x and obtain the domain.
• To calculate the range of the function, we simply express x as x=g(y) and then find the domain of g(y).

How do you find the range of a function?

Overall, the steps for algebraically finding the range of a function are:

• Write down y=f(x) and then solve the equation for x, giving something of the form x=g(y).
• Find the domain of g(y), and this will be the range of f(x).
• If you can't seem to solve for x, then try graphing the function to find the range.

How do you write the domain of a function?

• Identify the input values.
• Since there is an even root, exclude any real numbers that result in a negative number in the radicand.
• Set the radicand greater than or equal to zero and solve for x .
• The solution(s) are the domain of the function.

According to the question:

Given that: f (x) = $\frac{4-x}{x-4}$

We know that f(x) is defined if x-4 $\neq 0$

$\Rightarrow x \neq 4$

So, the domain of f(x) is = R- {4}

Let f(x) = y = $\frac{4-x}{x-4}$

$\begin{aligned}&\Rightarrow y x-4 y=4-x \\&\Rightarrow y x+x=4 y+4 \\&\Rightarrow x(y+1)=4 y+4 \\&\Rightarrow x=\frac{4(1+y)}{1+y}\end{aligned}$

If x is real number, then 1+y $\neq 0$

$\Rightarrow y \neq-1$

Therefore, Range of f(x)=R - {-1 }

Hence, The domain and range of the real function f defined by f(x)= $\frac{4-x}{x-4}$  is given by, Domain = $\mathbf{R}-\{4\}$, Range = $\{-1\} .$

Learn more about  domain and range here,

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