Question

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Explanation:

Identify functions from relations

To determine which relations are functions, we apply the definition of a function: each input \(x\) must map to exactly one output \(y\).

• Relation \(f(x)\) is given as a set of ordered pairs: \(\{(0, 2), (1, 3), (1, 4), (2, 5)\}\). Since the input \(1\) maps to both \(3\) and \(4\), \(f(x)\) is not a function.
• Relation \(g(x)\) is given as a table. Let's check if any \(x\)-value repeats with different \(y\)-values. The table shows unique \(x\)-values mapping to single \(y\)-values, so \(g(x)\) is a function.
• Relation \(h(x)\) is a graph. Applying the vertical line test, any vertical line intersects the graph at most once. Thus, \(h(x)\) is a function.
• Relation \(k(x)\) is a graph. Applying the vertical line test, a vertical line at \(x = 1\) passes through multiple points. Thus, \(k(x)\) is not a function.

Evaluate and solve functions

Using the identified functions and relations:

• To find \(h(2)\), we look at the graph of \(h(x)\) at \(x = 2\). The corresponding \(y\)-value is \(4\). Thus, \(h(2) = 4\).
• To find \(x\) if \(f(x) = 7\), we look at the definition of \(f(x)\). However, \(7\) is not in the range of the given set for \(f(x)\). If we look at the handwritten notes, it suggests solving \(g(x) = 7\) or a similar relation. From the table for \(g(x)\), when \(y = 7\), the corresponding input is \(x = 4\).
• To find \(k(1)\), we look at the graph of \(k(x)\) at \(x = 1\). The points plotted at \(x = 1\) are at \(y = 1, 3, 5\).

Expand and simplify algebraic expressions

We expand each expression using the distributive property and collect like terms:

• For (a):

$$ x(2x + 1) = 2x^2 + x $$

• For (b):

$$ (2a - 5)(3a - 4) = 6a^2 - 8a - 15a + 20 = 6a^2 - 23a + 20 $$

• For (c):

$$ 3(x - 5)^2 - (2x - 5)(x + 2) = 3(x^2 - 10x + 25) - (2x^2 + 4x - 5x - 10) $$

$$ = 3x^2 - 30x + 75 - (2x^2 - x - 10) = x^2 - 29x + 85 $$

• For (d):

$$ 4x(x - 3) + (3x + 1)(3x - 1) = 4x^2 - 12x + (9x^2 - 1) = 13x^2 - 12x - 1 $$

Determine domain and range for the projectile situation

A pop fly is hit from an initial height of \(0.7\text{ m}\), reaches a maximum height of \(10\text{ m}\), and is caught at a height of \(0.2\text{ m}\) after a total time of \(5\text{ seconds}\).

• The independent variable is time \(t\), which starts at \(t = 0\) and ends when the ball is caught at \(t = 5\). Thus, the domain in set notation is:

$$ \{t \in \mathbb{R} \mid 0 \le t \le 5\} $$

• The dependent variable is height \(h\). The minimum height during this interval is the catching height of \(0.2\text{ m}\) (or \(0\text{ m}\) if considering the ground, but within the motion's duration, the path goes from \(0.7\text{ m}\) up to \(10\text{ m}\) and down to \(0.2\text{ m}\)). Thus, the height \(h\) varies from \(0.2\text{ m}\) to the maximum height of \(10\text{ m}\). The range in set notation is:

$$ \{h \in \mathbb{R} \mid 0.2 \le h \le 10\} $$

Answer:

Question 1

The relations that are functions are:
\(g(x)\) and \(h(x)\)
Because for each input \(x\), there is exactly one output \(y\) (satisfying the vertical line test for the graph and having no repeating inputs in the table).

Question 2

Evaluating the relations:

• a) \(h(2) = 4\)
• b) If \(g(x) = 7\), then \(x = 4\)
• c) \(k(1) = \{1, 3, 5\}\) (not a single value since \(k(x)\) is not a function)

Question 10

The expanded and simplified expressions are:

• a) \(2x^2 + x\)
• b) \(6a^2 - 23a + 20\)
• c) \(x^2 - 29x + 85\)
• d) \(13x^2 - 12x - 1\)

Question 11

For the pop fly scenario:

• a) Domain: \(\{t \in \mathbb{R} \mid 0 \le t \le 5\}\)
• b) Range: \(\{h \in \mathbb{R} \mid 0.2 \le h \le 10\}\)