Mathematics Assignment: Exploring Functions, Graphs, and Inverses

This document presents solutions to a mathematics assignment, covering several key concepts in functions and algebra. The assignment includes questions on graphing functions, determining function values from graphs, and finding the inverse of a function. It explores transformations of functions, such as horizontal shifts, and solving equations involving functions. Specific problems include analyzing the graphs of given functions, finding the inverse of a linear function, and performing composite function operations. The solutions demonstrate step-by-step calculations and graphical representations to illustrate the concepts. This assignment provides a comprehensive overview of fundamental mathematical principles related to functions, equations, and their graphical interpretations.

Math Assignment
Name of the Student
Name of the University

Answer to question 1:
Given
Therefore,
X -∞ -6 0 2 3 4 5 ∞
g(x
) ∞ 10 13 13
1.6
9 4
7.8
1 ∞
Hence, the graph will be as follow:
Answer to question 2:
Given
Therefore,
X -4 -2 0 2 4

f(x
) 0
-
32 0 0
25
6
Hence, the graph will look like:
Answer to question 3:
Given
f ( x )=−2 x2−x+ 15
x2−x−12
Therefore, the graph will look like

Answer to question 4:
Given,
f ( x ) =5 x +8
Let, f(x) = y
Hence,
Y = 5x+8
Or, x = (y-8)/5
Hence, f-1(x) = (x-8)/5
Therefore,
f-1(f(x)) = (f(x)-8)/5
= (5x+8-8)/5
=5x/5
=x
Hence, (f−1 ° f ¿ ( x )=x
Answer to question 5:
Given, g(x) = x^2 + 3

The graph is shown in purple color in the following figure.
Now, considering the horizontal transformation at x = -2, -1 ,1, 2
We have
g(x-2) = (x-2)^2 + 3 shown in black color
g(x-1) = (x-1)^2 +3 shown in red color
g(x+1) = (x+1)^2 +3 shown sky blue color
g(x+2) = (x+2)^2 + 3 shown in green color
Answer to question 6:
Given
√ x+5+ 1= 6
√ x +5
Multiplying both side by √ x+5
We have,
(√ x+5 ¿∗( √ x +5+1 )=6
Or, (x+5)+ √ x+5 = 6

Or, x+5 = (6-x-5)^2
Or, x +5 = (1-x)^2
Or, x^2 -2x+1-x-5 = 0
Or, x^2-3x-4 =0
Or, x^2 -4x+x-4=0
Or, (X-4)(x+1) =0
Therefore, x = 4, -1
Answer to question 7:
Given that f ( x )= √x and g ( x ) = x −3
2
Let, y = g(x)
Hence, y = (x-3)/2
Or, x = 2y+3
Hence, g-1(x) = 2x+3
Now, f(g-1(x)) = √(2 x +3)
Again,
( g−1 ∘ g−1 ) ( x )=2 ( 2 x+ 3 ) +3
= 4x+9
Answer to question 8:
Given that f ( x )=3 x +1 , g ( x )=x2−2 x−6 ,∧h ( x )=x3
Hence, h(g(x)) = (x2-2x-6)^3
=x^6-6x^5-6x^4+64x^3+36x^2-216x-216
Therefore,
f(h(g(x))) = 3(x^6-6x^5-6x^4+64x^3+36x^2-216x-216)+1
=3x^6-18x^5 – 18 x^4 + 192 x ^3 + 108 x^2 -648x – 647
Again,
(f ∘ g)(−1)
First of all, f(g(x)) =3(x^2-2x-6)+3
= 3x^2-6x-15

Therefore, f(g(-1))
= 3(-1)^2-6(-1)-15
= 3+6-15
=-6