Answer:
Step-by-step explanation:
Function A:
(
)
=
2
−
2
F(x)=2x−2
1. Finding the X-Intercept
The x-intercept is where the graph crosses the x-axis. This happens when
=
0
y=0.
Set
(
)
=
0
F(x)=0:
2
−
2
=
0
2x−2=0
2
=
2
2x=2
=
1
x=1
So, the x-intercept is
(
1
,
0
)
(1,0).
2. Finding the Y-Intercept
The y-intercept is where the graph crosses the y-axis. This happens when
=
0
x=0.
Set
=
0
x=0:
(
0
)
=
2
(
0
)
−
2
=
−
2
F(0)=2(0)−2=−2
So, the y-intercept is
(
0
,
−
2
)
(0,−2).
3. Domain and Range
The domain of
(
)
=
2
−
2
F(x)=2x−2 is all real numbers,
R, because there are no restrictions on
x.
The range is also all real numbers,
R, because as
x takes any value,
(
)
F(x) will take any value.
4. Sketching the Graph
The graph of
(
)
=
2
−
2
F(x)=2x−2 is a straight line with a slope of 2 and a y-intercept at
(
0
,
−
2
)
(0,−2).
Function B:
(
)
=
109
(
+
2
)
F(x)=109a(x+2)
To analyze this function, we need to understand
a and its effect. Let's assume
a is a constant and
109
109a represents the slope.
1. Finding the X-Intercept
Set
(
)
=
0
F(x)=0:
109
(
+
2
)
=
0
109a(x+2)=0
+
2
=
0
x+2=0
=
−
2
x=−2
So, the x-intercept is
(
−
2
,
0
)
(−2,0).
2. Finding the Y-Intercept
Set
=
0
x=0:
(
0
)
=
109
(
0
+
2
)
=
218
F(0)=109a(0+2)=218a
So, the y-intercept is
(
0
,
218
)
(0,218a).
3. Domain and Range
The domain of
(
)
=
109
(
+
2
)
F(x)=109a(x+2) is all real numbers,
R, because there are no restrictions on
x.
The range is also all real numbers,
R, because as
x takes any value,
(
)
F(x) will take any value depending on
a.
4. Sketching the Graph
The graph of
(
)
=
109
(
+
2
)
F(x)=109a(x+2) is a straight line with a slope of
109
109a and a y-intercept at
(
0
,
218
)
(0,218a).
Graphs
To sketch the graphs, we plot the intercepts and draw the lines:
Function A:
(
)
=
2
−
2
F(x)=2x−2
X-intercept:
(
1
,
0
)
(1,0)
Y-intercept:
(
0
,
−
2
)
(0,−2)
Function B:
(
)
=
109
(
+
2
)
F(x)=109a(x+2)
X-intercept:
(
−
2
,
0
)
(−2,0)
Y-intercept:
(
0
,
218
)
(0,218a)
Let's plot these graphs for visual understanding.
I'll provide the sketches of these graphs using Python.