Question

Having A really hard time and my Professor is being a A Hole with help so can someone just solve it for me I need the range and the graph.​

Answer 1

Normally, when $x=3$ the function

$f(x)=\dfrac12(x-1)^3-1$

takes on a value of

$f(3)=\dfrac12(3-1)^3-1=\dfrac82-1=4-1=3$

so the graph of $f(x)$ passes through the point (3, 3). This is the only place at which the cubic takes on a value of 3, since $f(x)$ is one-to-one.

In the definition of $q(x)$, however, we remove this point from the curve and replace with the point (3, -2).

While $f(x)$ is a standard cubic function, so that its range is all real numbers, the replacement made for $q(x)$ removes 3 from the range set, so that the range of $q(x)$ would be all real numbers *except* 3.