For the real-valued functions g(x)=(x-4)/(x-5) and h(x)=4x+1 . find the composition gcirc h and specify its domain using interval notation. (gcirc h)(x)=square Domain of gcdot h:square

To find the composition $$(g \circ h)(x)$$ , we need to substitute $$h(x)$$ into $$g(x)$$ .

Given:
$$g(x) = \frac{x-4}{x-5}$$
$$h(x) = 4x+1$$

Substituting $$h(x)$$ into $$g(x)$$ :
$$(g \circ h)(x) = g(h(x)) = g(4x+1) = \frac{(4x+1)-4}{(4x+1)-5} = \frac{4x-3}{4x-4}$$

The domain of $$(g \circ h)(x)$$ is the set of all real numbers except for the values that make the denominator zero. In this case, the denominator is $$(4x-4)$$ , which is zero when $$x = 1$$ . Therefore, the domain of $$(g \circ h)(x)$$ is all real numbers except $$x = 1$$ .

In interval notation, the domain of $$(g \circ h)(x)$$ is $$(-\infty, 1) \cup (1, \infty)$$ .