Question

$f$ is a quadratic equation whose graph is a parabola opening upward and has a vertex on the x-axis. The graph of the new function $g$ defined by $g\left(x\right)=2-f(x-5)$ has a range defined by the interval

• A. $[-5,+\infty )$
• B. $[2,3)$
• C. $(-\infty ,2]$
• D. None of these

Answer 1

Answer: (C)

$(-\infty ,2]$

Since $f$ is a quadratic equation it will have variable of degree $2$ and its graph is a parabola. So, it will be in the form of $f\left(x\right)={(x-a)}^2$ and it lies between $0$ to $\infty$.

Now, $f(x-5)={(x-(a+5))}^2$ is shifted to the right. So, range won't change.

$-f(x-5)$ lies ranges from $-\infty$ to $0$.

$g\left(x\right)=2-f(x-5)$ shifted $2$ units upward.

$\therefore$ Range of $g\left(x\right)=2-f(x-5)$ is $(-\infty ,2]$.