Question

problems 12 - 13, use the graph of each function to answer the question set. 12. the graph of y = f(x), consisting of two line segments and a semicircle, is shown for -4 ≤ x ≤ 5. let g(x)=-f(x - 3)+2 a. graph g(x) b. state the domain and range of g. c. find g(2) d. find zeros of g(x) e. state the y - intercept of g.

Explanation:

Step1: Analyze function transformation

The transformation $g(x)=-f(x - 3)+2$ involves a horizontal shift of 3 units to the right, a reflection about the x - axis, and a vertical shift of 2 units up of the function $y = f(x)$.

Step2: Find domain of $g(x)$

Given the domain of $f(x)$ is $-4\leq x\leq5$. For $g(x)=-f(x - 3)+2$, we set $-4\leq x - 3\leq5$. Solving for $x$ gives $- 1\leq x\leq8$. So the domain of $g(x)$ is $[-1,8]$.

Step3: Find range of $g(x)$

The range of $f(x)$ needs to be considered with the transformations. First, reflecting about the x - axis and then shifting up by 2. If the range of $f(x)$ is say $[a,b]$, the range of $-f(x)$ is $[-b,-a]$ and the range of $-f(x)+2$ is $[-b + 2,-a+2]$.

Step4: Calculate $g(2)$

$g(2)=-f(2 - 3)+2=-f(-1)+2$. We find the value of $f(-1)$ from the graph of $y = f(x)$ and then calculate $g(2)$.

Step5: Find zeros of $g(x)$

Set $g(x)=0$, so $-f(x - 3)+2 = 0$, which gives $f(x - 3)=2$. We find the x - values of $f(x)$ for which $y = 2$ from the graph of $f(x)$ and then solve $x-3$ equal to those x - values for $f(x)$ to get the zeros of $g(x)$.

Step6: Find y - intercept of $g(x)$

The y - intercept of $g(x)$ is $g(0)=-f(0 - 3)+2=-f(-3)+2$. We find $f(-3)$ from the graph of $y = f(x)$ and then calculate $g(0)$.

Answer:

A. Graph $g(x)$ by applying the described transformations to the graph of $f(x)$.
B. Domain: $[-1,8]$, Range: (needs values from $f(x)$ range after transformation)
C. Calculate $g(2)=-f(-1)+2$ (value depends on $f(-1)$ from graph)
D. Solve $f(x - 3)=2$ for $x$ using the graph of $f(x)$ to find zeros.
E. Calculate $g(0)=-f(-3)+2$ (value depends on $f(-3)$ from graph)