### 4.1 Examination
#### 4.1.1 Relationship Type (1 Mark)
The given relationship from "Race" to "Food Eaten" is such that each race has one specific type of food associated with it. Therefore, the kind of relationship that exists from "Race" to "Food Eaten" is a Function.
#### 4.1.2 Is it a Function? (2 Marks)
Yes, this relationship is a function. A function is defined as a relation in which every element of the domain (in this case, "Race") is associated with exactly one element of the codomain (in this case, "Food Eaten"). Here, each race (Chinese, Whites, Africans) is linked to one unique type of food (Frogs, Nomdlomboyi, Shrimp), fulfilling the criteria for a function.
### 4.2 Consider the two functions defined by [tex]\( f(x) = x^2 - 4 \)[/tex] and [tex]\( g(x) = 2^x \)[/tex]
#### 4.2.1 Find the value of [tex]\( f(4) \)[/tex] (1 Mark)
To find [tex]\( f(4) \)[/tex]:
[tex]\[ f(x) = x^2 - 4 \][/tex]
Substitute [tex]\( x = 4 \)[/tex]:
[tex]\[ f(4) = 4^2 - 4 = 16 - 4 = 12 \][/tex]
The value of [tex]\( f(4) \)[/tex] is 12.
#### 4.2.2 Write down the x-intercepts of [tex]\( f \)[/tex] (2 Marks)
The x-intercepts occur where [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ x^2 - 4 = 0 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x^2 = 4 \][/tex]
[tex]\[ x = \pm 2 \][/tex]
The x-intercepts are -2 and 2.
#### 4.2.3 Determine the coordinates of the y-intercept of [tex]\( g \)[/tex] (2 Marks)
The y-intercept occurs where [tex]\( x = 0 \)[/tex]:
[tex]\[ g(x) = 2^x \][/tex]
Substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = 2^0 = 1 \][/tex]
The coordinates of the y-intercept are [tex]\((0, 1)\)[/tex].
#### 4.2.4 Determine the equation of an asymptote of [tex]\( g \)[/tex] (1 Mark)
For the function [tex]\( g(x) = 2^x \)[/tex], as [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( g(x) \rightarrow 0 \)[/tex]. Therefore, the horizontal asymptote is:
[tex]\[ y = 0 \][/tex]
The equation of the asymptote is [tex]\( y = 0 \)[/tex].
#### 4.2.5 Write the coordinates of the turning point of [tex]\( f \)[/tex] (2 Marks)
The function [tex]\( f(x) = x^2 - 4 \)[/tex] is a parabola opening upwards. The vertex (turning point) of this parabola is found at the minimum point. For the quadratic [tex]\( ax^2 + bx + c \)[/tex], the vertex is at [tex]\( x = -\frac{b}{2a} \)[/tex].
For [tex]\( f(x) = x^2 - 4 \)[/tex]:
[tex]\[ a = 1, \quad b = 0, \quad c = -4 \][/tex]
The x-coordinate of the vertex is:
[tex]\[ x = -\frac{0}{2(1)} = 0 \][/tex]
Substitute [tex]\( x = 0 \)[/tex] back into [tex]\( f(x) \)[/tex]:
[tex]\[ f(0) = 0^2 - 4 = -4 \][/tex]
The coordinates of the turning point are [tex]\((0, -4)\)[/tex].
