Question

1. the graph of (y = f(x)), consisting of two - line segments and a semi - circle, is shown for (-4leq xleq3). sketch a graph of (m) on the same axes above where (m(x)=f(2x)).
2. the domain of a function (h) is (-4leq xleq7) and the range of (h) is (-6leq yleq0). find the domain and range of (g), where (g(x)=3h(x - 2)).
3. the domain of a function (k) is (2leq xleq14) and the range of (k) is (-3leq yleq2). find the domain and range of (r), where (r(x)=-2k(2x)).
4. the domain of a function (f) is (-6leq xleq4) and the range of (f) is (-10leq yleq3). find the domain and range of (p), where (p(x)=5 - 3f(2(x + 1))).

worksheet a: transformations of functions
topic 1.12
created by bryan passwater

Explanation:

12.

Step1: Recall horizontal - compression rule

For a function $y = f(x)$ and $y=f(bx)$ where $b>0$, the graph of $y = f(bx)$ is a horizontal compression of the graph of $y = f(x)$ by a factor of $\frac{1}{b}$. Here $b = 2$, so the graph of $m(x)=f(2x)$ is a horizontal compression of the graph of $y = f(x)$ by a factor of $\frac{1}{2}$.

Step2: Transform key - points

If a point $(x,y)$ lies on $y = f(x)$, then the point $(\frac{x}{2},y)$ lies on $y = f(2x)$. For example, if $f(x)$ has a point $(-4,y_1)$ on its graph, then $m(x)$ has a point $(- 2,y_1)$ on its graph. Repeat this process for all key - points (intersection points with axes, endpoints of segments, etc.) on the graph of $f(x)$ and then sketch the graph of $m(x)$.

13.

Step1: Find the domain of $g(x)$

For the function $g(x)=3h(x - 2)$, we use the domain of $h(x)$. Let $u=x - 2$. Since the domain of $h(u)$ is $-4\leq u\leq7$, we set $-4\leq x - 2\leq7$.
Adding 2 to all parts of the inequality: $-4+2\leq x-2 + 2\leq7 + 2$, so $-2\leq x\leq9$.

Step2: Find the range of $g(x)$

The range of $h(x)$ is $-6\leq y\leq0$. For $g(x)=3h(x - 2)$, we multiply the range of $h(x)$ by 3. If $y_h$ is the value of $h(x)$ and $y_g$ is the value of $g(x)$, then $y_g = 3y_h$. So $3\times(-6)\leq3y_h\leq3\times0$, and the range of $g(x)$ is $-18\leq y\leq0$.

Step1: Find the domain of $r(x)$

For the function $r(x)=-2k(2x)$, let $u = 2x$. Since the domain of $k(u)$ is $2\leq u\leq14$, we set $2\leq2x\leq14$.
Dividing all parts of the inequality by 2: $\frac{2}{2}\leq\frac{2x}{2}\leq\frac{14}{2}$, so $1\leq x\leq7$.

Step2: Find the range of $r(x)$

The range of $k(x)$ is $-3\leq y\leq2$. For $r(x)=-2k(2x)$, if $y_k$ is the value of $k(x)$ and $y_r$ is the value of $r(x)$, then $y_r=-2y_k$.
Multiply the range of $k(x)$ by - 2. When we multiply an inequality by a negative number, the direction of the inequality sign changes. So $-2\times2\leq-2y_k\leq-2\times(-3)$, and the range of $r(x)$ is $-4\leq y\leq6$.

Answer:

Domain: $[-2,9]$, Range: $[-18,0]$

14.