Question

Let f(x) be a function with domain [−3,∞) and range (−∞,2). If g(x)= f^−1(x) what is the domain of g(x+4)−7?

Answer 1

Answer:

• the domain of the function $g(x+4)-7$ is $(-\infty,-2);$

• the range of the function $g(x+4)-7$ is $[-10,\infty).$

Step-by-step explanation:

If the function $f(x)$ has the domain $[-3,\infty)$ and the range $(-\infty,2),$ then the function $g(x)=f^{-1}(x)$ has the domain that id the function's $f(x)$ range and the range that is function's $f(x)$ domain. Therefore,

• the domain of the function $g(x)$ is $(-\infty,2);$

• the range of the function $g(x)$ is $[-3,\infty).$

The function $g(x+4)-7$ is the translation of the function $g(x)$ 4 units to the left and 7 units down, hence

• the domain of the function $g(x+4)-7$ is $(-\infty,-2);$

• the range of the function $g(x+4)-7$ is $[-10,\infty).$