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MAT 153: College Algebra and Trigonometry

Section 1.1 Representing Mathematical Relationships

Mathematical relationships between two quantities can be represented in multiple ways: as a table, as a graph or as an equation. Each of these representations adds to our understanding of a mathematical situation, and we should be able to approach a given situation by considering each of these representations.
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This diagram showing four interconnected representations of a mathematical relationship: Graph, Table, Equation, and Verbal Description. The Graph is at the top, with arrows pointing to both Table and Equation. Table and Equation are connected by a double arrow. The label Verbal Description appears in the center, indicating connections among all three representations.🔗
Figure 1.1.1.
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The domain of a function is the set of inputs to the function. The range of a function is the set of outputs from the function.
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A decreasing graph showing the amount of punch in a punchbowl after t hours.
Figure 1.1.2.
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Table 1.1.3. Table for \(V(t)\)
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\(t\) (hours) \(V\) (inches3)
0 4189
0.5 3934
1 3284
1.5 2408
2 1474

Investigation 1.1.1. Punch Bowl Investigation.

Table 1.1.3 and Figure 1.1.2 show the relationship between \(V\text{,}\) the volume of punch in a bowl at a party, and \(t\text{,}\) the time in hours since the party began.
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Use the table and graph above for \(V(t)\) to answer the following questions:
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  1. Add two additional points to the table by using information contained in the graph.
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  2. What is the volume of punch in the bowl 1 hour after the party started?
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  3. When was the volume of the punch approximately 650 in\(^3\text{?}\)
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  4. What was the volume of punch in the bowl when the party started?
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  5. When will the volume of punch in the bowl be half of the volume when the party started?
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  6. When will the party run out of punch?
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  7. In what time interval will the volume of punch in the bowl be between 1474 in\(^3\) and 3284 in\(^3\text{?}\)
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  8. When is the volume of punch in the bowl decreasing fastest? When is the volume of the punch in the bowl decreasing slowest?
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  9. What are the domain and range of \(V\text{?}\)
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  10. If the volume of punch in the bowl is given by the equation
    \begin{equation*} V(t)=72\pi t^3-360 \pi t^2+\frac{4000}{3} \pi \end{equation*}
    what is the volume of punch in the bowl 1.25 hours after the party started? Add this information to the graph and table above.
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Problem 1.1.4. Hiker Elevation.

The graph below shows the elevation of a hiker, \(E(t)\) (in feet), \(t\) hours into a 3 hour hike.
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Graph of E of t showing a wavelike pattern that oscillates between about 1000 and 1500 as t increases from 0 to about 3. The curve rises quickly near t equals 0, then fluctuates with small peaks and dips before decreasing slightly after t equals 3.
Figure 1.1.5.
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  1. What is the elevation of the hiker after 1.5 hours?
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  2. At what times is the hiker at an elevation of 1200 feet?
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  3. Solve the equation \(E(t)=1200\text{.}\)
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  4. At what times is the hiker at an elevation of at least 1200 feet?
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  5. Solve the inequality \(E(t) \geq 1200\text{.}\)
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  6. What is the maximum elevation of the hiker? At what time is he at this elevation?
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A function is a specific kind of relation, where each input of the function leads to only one output. Functions are used throughout math and science, and will be a primary focus in this course.
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Example 1.1.6. Using the graph of \(g\).

The function \(g\) is graphed below. Use the graph to answer the following questions.
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A piecewise graph of g(x) appears on a coordinate grid. It begins with a solid point at (-4, 2) connected by a decreasing line to an open circle at (-2, 0.5). From that open circle, the graph continues downward to a solid point at (-1, 0). A rising line segment connects (-1, 0) to a solid point at (2, 2). Farther right, there is an open circle at (4, -1), and from that point the graph becomes a smooth upward-curving parabola ending at a solid point at (5, 3). Open circles indicate values where the function is not defined, and solid circles show defined points.
Figure 1.1.7.
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  1. Evaluate:
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    1. \(\displaystyle g(0)\)
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    2. \(\displaystyle g(2)\)
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    3. \(\displaystyle g(-2)\)
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  2. Solve:
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    1. \(\displaystyle g(x) = -1\)
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    2. \(\displaystyle g(x) = 3\)
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    3. \(\displaystyle g(x) = 0\)
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    4. \(\displaystyle g(x) < 0\)
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Problem 1.1.8.

The graph of \(g\) is shown below.
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Graph of g of x showing a cubic-shaped curve. The graph rises from the lower left, crosses the x-axis near x equals negative four, reaches a local maximum at approximately negative three point six eight with y about three point four, then decreases, crossing the x-axis again at negative one and zero. It continues downward to a local minimum at about x equals one point four with y about negative nine point six, and then rises steeply through x equals three where g of x equals zero.
Figure 1.1.9.
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  1. Evaluate \(g(-3.5)\text{.}\)
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  2. Solve \(g(x) = 0\text{.}\)
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  3. Solve \(g(x) > 0\text{.}\)
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  4. Solve \(g(x) \leq 0\text{.}\)
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Problem 1.1.10.

Use Desmos to graph the function \(g(x)=x+\sin(2x)\text{.}\)
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  1. Evaluate \(g(0)\text{.}\)
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  2. Evaluate \(g\!\left(- \frac{\pi}{3}\right)\text{.}\)
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  3. Solve \(g(x)=1.288\text{.}\)
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  4. Solve \(g(x)=10\text{.}\)
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  5. Solve \(g(x)=-1200\text{.}\)
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Definition 1.1.11.

To determine the output of a function for a given input using an equation, substitute the given input into the equation. Evaluate the resulting expression making sure to follow the order of operations:
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  1. Simplify expressions within grouping symbols (parentheses, roots and fraction bars)
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  2. Evaluate exponents
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  3. Perform multiplication and division from left to right
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  4. Perform addition and subtraction from left to right
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Problem 1.1.12.

Evaluate each function for the indicated value or expression.
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  1. \(\displaystyle k(t) = 3t^3 - 2t^2 + t,\; k(-2)\)
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  2. \(\displaystyle q(x) = \dfrac{x - 2}{x^2 + 2},\; q(n^2)\)
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  3. \(\displaystyle f(n) = 2 \cdot 3^{2n - 3},\; f\!\left(\tfrac{1}{2}\right)\)
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  4. \(\displaystyle m(x) = \sqrt{x - 3} + 4,\; m(x^2 + 4)\)
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