IDNLearn.com provides a user-friendly platform for finding and sharing accurate answers. Our platform is designed to provide quick and accurate answers to any questions you may have.
Sagot :
To determine the domain of the function [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] where [tex]\(f(x) = x^2 - 25\)[/tex] and [tex]\(g(x) = x - 5\)[/tex], we need to find the values of [tex]\(x\)[/tex] for which the function is defined.
1. Consider the denominator [tex]\(g(x)\)[/tex] of the function first:
[tex]\[ g(x) = x - 5 \][/tex]
For the function [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] to be defined, the denominator [tex]\(g(x)\)[/tex] should not be zero. Setting the denominator equal to zero to find the critical points:
[tex]\[ x - 5 = 0 \][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[ x = 5 \][/tex]
So, the function [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] is not defined at [tex]\(x = 5\)[/tex].
2. Consider the numerator [tex]\(f(x)\)[/tex]:
[tex]\[ f(x) = x^2 - 25 \][/tex]
Factor it to better identify any restrictions:
[tex]\[ f(x) = (x + 5)(x - 5) \][/tex]
The factored form shows that [tex]\(f(x)\)[/tex] has zeros at:
[tex]\[ x = 5 \quad \text{and} \quad x = -5 \][/tex]
However, the zeros of [tex]\(f(x)\)[/tex] do not affect the domain of [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] except through [tex]\(g(x)\)[/tex].
3. Determine the actual restrictions on the domain:
- The only value that makes the denominator zero is [tex]\(x = 5\)[/tex].
Hence, the domain of the function [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] is all real values of [tex]\(x\)[/tex] except where [tex]\(x = 5\)[/tex].
So, the correct answer is:
[tex]\[ \text{all real values of } x \text{ except } x=5 \][/tex]
1. Consider the denominator [tex]\(g(x)\)[/tex] of the function first:
[tex]\[ g(x) = x - 5 \][/tex]
For the function [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] to be defined, the denominator [tex]\(g(x)\)[/tex] should not be zero. Setting the denominator equal to zero to find the critical points:
[tex]\[ x - 5 = 0 \][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[ x = 5 \][/tex]
So, the function [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] is not defined at [tex]\(x = 5\)[/tex].
2. Consider the numerator [tex]\(f(x)\)[/tex]:
[tex]\[ f(x) = x^2 - 25 \][/tex]
Factor it to better identify any restrictions:
[tex]\[ f(x) = (x + 5)(x - 5) \][/tex]
The factored form shows that [tex]\(f(x)\)[/tex] has zeros at:
[tex]\[ x = 5 \quad \text{and} \quad x = -5 \][/tex]
However, the zeros of [tex]\(f(x)\)[/tex] do not affect the domain of [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] except through [tex]\(g(x)\)[/tex].
3. Determine the actual restrictions on the domain:
- The only value that makes the denominator zero is [tex]\(x = 5\)[/tex].
Hence, the domain of the function [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] is all real values of [tex]\(x\)[/tex] except where [tex]\(x = 5\)[/tex].
So, the correct answer is:
[tex]\[ \text{all real values of } x \text{ except } x=5 \][/tex]
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. IDNLearn.com has the solutions you’re looking for. Thanks for visiting, and see you next time for more reliable information.