Answer :
Let's break down the problem and solve each part step-by-step:
### PART 4.1
4.1.1 Write down the kind of relationship that exists from Race to Food Eaten
To establish the kind of relationship that exists from Race to Food Eaten, we need to understand the nature of how these sets interact. Since each race can eat multiple types of food, but a given type of food may be eaten by more than one race, the relationship from Race to Food Eaten is a "one-to-many" relationship.
4.1.2 Is this relationship a function or not? Explain your answer.
A function in mathematics is defined as a relationship where each input is associated with exactly one output. In this case, if a race can eat multiple kinds of food, then a single race (input) maps to multiple food types (outputs). Therefore, the relationship from Race to Food Eaten is not a function because it does not fulfill the requirement of each input having exactly one output.
### PART 4.2
4.2.1 Find the value of [tex]\(f(4)\)[/tex]
Given the function [tex]\( f(x) = x^2 - 4 \)[/tex]:
[tex]\[ f(4) = 4^2 - 4 = 16 - 4 = 12 \][/tex]
So, the value of [tex]\( f(4) \)[/tex] is [tex]\( 12 \)[/tex].
4.2.2 Write down the [tex]\(x\)[/tex]-intercepts of [tex]\(f\)[/tex]
To find the [tex]\( x \)[/tex]-intercepts of [tex]\( f(x) = x^2 - 4 \)[/tex], we need to solve for [tex]\( x \)[/tex] when [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ x^2 - 4 = 0 \][/tex]
[tex]\[ (x - 2)(x + 2) = 0 \][/tex]
So, [tex]\( x = 2 \)[/tex] and [tex]\( x = -2 \)[/tex].
The [tex]\( x \)[/tex]-intercepts are [tex]\( x = 2 \)[/tex] and [tex]\( x = -2 \)[/tex].
4.2.3 Determine the coordinates of the [tex]\( y \)[/tex]-intercept of [tex]\( g \)[/tex]
Given the function [tex]\( g(x) = 2^x \)[/tex], to find the [tex]\( y \)[/tex]-intercept, we set [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = 2^0 = 1 \][/tex]
The coordinates of the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex] are [tex]\( (0, 1) \)[/tex].
4.2.4 Determine the equation of an asymptote of [tex]\( g \)[/tex]
The function [tex]\( g(x) = 2^x \)[/tex] is an exponential function. As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( g(x) \)[/tex] approaches but never reaches 0.
Therefore, the horizontal asymptote of [tex]\( g(x) \)[/tex] is:
[tex]\[ y = 0 \][/tex]
### Summary
- 4.1.1: Relationship is "one-to-many".
- 4.1.2: The relationship is not a function because one race can map to multiple foods.
- 4.2.1: [tex]\( f(4) = 12 \)[/tex]
- 4.2.2: The [tex]\( x \)[/tex]-intercepts are [tex]\( x = 2 \)[/tex] and [tex]\( x = -2 \)[/tex].
- 4.2.3: The coordinates of the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex] are [tex]\( (0, 1) \)[/tex].
- 4.2.4: The equation of the asymptote of [tex]\( g(x) \)[/tex] is [tex]\( y = 0 \)[/tex].
### PART 4.1
4.1.1 Write down the kind of relationship that exists from Race to Food Eaten
To establish the kind of relationship that exists from Race to Food Eaten, we need to understand the nature of how these sets interact. Since each race can eat multiple types of food, but a given type of food may be eaten by more than one race, the relationship from Race to Food Eaten is a "one-to-many" relationship.
4.1.2 Is this relationship a function or not? Explain your answer.
A function in mathematics is defined as a relationship where each input is associated with exactly one output. In this case, if a race can eat multiple kinds of food, then a single race (input) maps to multiple food types (outputs). Therefore, the relationship from Race to Food Eaten is not a function because it does not fulfill the requirement of each input having exactly one output.
### PART 4.2
4.2.1 Find the value of [tex]\(f(4)\)[/tex]
Given the function [tex]\( f(x) = x^2 - 4 \)[/tex]:
[tex]\[ f(4) = 4^2 - 4 = 16 - 4 = 12 \][/tex]
So, the value of [tex]\( f(4) \)[/tex] is [tex]\( 12 \)[/tex].
4.2.2 Write down the [tex]\(x\)[/tex]-intercepts of [tex]\(f\)[/tex]
To find the [tex]\( x \)[/tex]-intercepts of [tex]\( f(x) = x^2 - 4 \)[/tex], we need to solve for [tex]\( x \)[/tex] when [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ x^2 - 4 = 0 \][/tex]
[tex]\[ (x - 2)(x + 2) = 0 \][/tex]
So, [tex]\( x = 2 \)[/tex] and [tex]\( x = -2 \)[/tex].
The [tex]\( x \)[/tex]-intercepts are [tex]\( x = 2 \)[/tex] and [tex]\( x = -2 \)[/tex].
4.2.3 Determine the coordinates of the [tex]\( y \)[/tex]-intercept of [tex]\( g \)[/tex]
Given the function [tex]\( g(x) = 2^x \)[/tex], to find the [tex]\( y \)[/tex]-intercept, we set [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = 2^0 = 1 \][/tex]
The coordinates of the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex] are [tex]\( (0, 1) \)[/tex].
4.2.4 Determine the equation of an asymptote of [tex]\( g \)[/tex]
The function [tex]\( g(x) = 2^x \)[/tex] is an exponential function. As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( g(x) \)[/tex] approaches but never reaches 0.
Therefore, the horizontal asymptote of [tex]\( g(x) \)[/tex] is:
[tex]\[ y = 0 \][/tex]
### Summary
- 4.1.1: Relationship is "one-to-many".
- 4.1.2: The relationship is not a function because one race can map to multiple foods.
- 4.2.1: [tex]\( f(4) = 12 \)[/tex]
- 4.2.2: The [tex]\( x \)[/tex]-intercepts are [tex]\( x = 2 \)[/tex] and [tex]\( x = -2 \)[/tex].
- 4.2.3: The coordinates of the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex] are [tex]\( (0, 1) \)[/tex].
- 4.2.4: The equation of the asymptote of [tex]\( g(x) \)[/tex] is [tex]\( y = 0 \)[/tex].
