Find the domain of the following functions: a) f(x) = (3x−2)/((x−3)(x+2)(x−1)) b) f(x) = (4x−3)/(5x²−5x) c)…

Question

Find the domain of the following functions: a) f(x) = (3x−2)/((x−3)(x+2)(x−1)) b) f(x) = (4x−3)/(5x²−5x) c) f(x) = (5x+1)/(4x²−9) d) f(x) = sqrt(x²−49) e) f(x) = sqrt(25-x²)

ScanSolve AI Tutor · Accepted Answer

To find the domain of a rational function, the denominator cannot be equal to zero.

Identify the denominator of the function.
The given function is $f(x) = \frac{5x+1}{4x^2-9}$ .
The denominator is $4x^2-9$ .
Set the denominator equal to zero to find the values of $x$ that are excluded from the domain.
$4x^2 - 9 = 0$
Solve for $x$ .
This is a difference of squares, $a^2 - b^2 = (a-b)(a+b)$ , where $a=2x$ and $b=3$ .
$(2x)^2 - 3^2 = 0$ $(2x-3)(2x+3) = 0$
Set each factor equal to zero:
$2x-3 = 0 \implies 2x = 3 \implies x = \frac{3}{2}$ $2x+3 = 0 \implies 2x = -3 \implies x = -\frac{3}{2}$
These are the values of $x$ for which the function is undefined.
State the domain.
The domain consists of all real numbers except for these values.
The domain can be written in set-builder notation as:
$\left\{x \in \mathbb{R} \mid x \neq -\frac{3}{2}, x \neq \frac{3}{2}\right\}$
The final answer is $\boxed{\left\{x \in \mathbb{R} \mid x \neq -\frac{3}{2}, x \neq \frac{3}{2}\right\}}$ .

Find the domain of the following functions: a) f(x) = (3x−2)/((x-3)(x+2)(x-1)) b) f(x) = (4x−3)/(5x²-5x) c) f(x) = (5x+1)/(4x²-9) d) f(x) = sqrt(x²−49) e) f(x) = sqrt(25−x²)

Step-by-step solution

Identify the denominator of the function.

Set the denominator equal to zero to find the values of $x$ that are excluded from the domain.

Solve for $x$ .

State the domain.

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