Given: 𝑓(𝑥) = 3 + 1 and h(𝑥) =− 1𝑥+5 𝑥−2 2

Question

4.1
Write down the equation of the asymptotes of 𝑓.
(2)
4.2
Determine the 𝑥 −intercepts of 𝑓.
(3)
4.3
Determine the 𝑦 −intercepts of 𝑓.
(2)
4.4
Sketch the graph of 𝑓 and h on the same set of axes, clearly indicating the distinctive features of the functions.
(5)
4.5
Determine the values 𝑥 for which 𝑓(𝑥) < h(𝑥).
(2)
4.6
Theline𝑦 =− 3 𝑥 + 25 cuts𝑓at𝐸(3;4)and𝐹.Writedownthe 44
coordinates of 𝐹.
(7)

Answer 1

4.1 Equation of the asymptotes of f:
The equation of the horizontal asymptote is y = 3.

4.2 x-intercepts of f:
To find the x-intercepts, set f(x) = 0:
3 + 1 = 0
4 = 0
There are no x-intercepts for f.

4.3 y-intercepts of f:
To find the y-intercepts, set x = 0 in f(x):
f(0) = 3(0) + 1
f(0) = 1
The y-intercept is at (0, 1).

4.4 Graph of f and h on the same set of axes:

The graph will show f(x) as a horizontal line at y = 3 with a y-intercept at (0, 1). The graph of h(x) will be a slanted line that intersects f at point E(3, 4). Identify the distinctive features of the functions clearly on the graph.

4.5 Values of x for which f(x) < h(x):
To find where f(x) < h(x), set the two functions equal to each other and solve for x:
3 + 1 < -1x+5 / x-2
4 < -1x+5 / x-2
4 < -1x+5 / x-2
Multiply both sides by x-2:
4x-8 < -x+5
Combine like terms:
5x < 13
x < 13/5
Therefore, f(x) < h(x) for x < 13/5.

4.6 Coordinates of F when y = -3x + 25 intersects f at E(3, 4):
Substitute the x-coordinate of point E into the equation y = -3x + 25:
y = -3(3) + 25
y = -9 + 25
y = 16
Therefore, the coordinates of F are (3, 16).