Problem 2 - Discrete Mathematics with Applications

Question

The Celsius and Fahrenheit scales are related by the formula \(\mathrm{F}=\frac{9}{5} \mathrm{C}+32\) $$ g(x)=\left\\{\begin{array}{ll}{2|x|+3} & {\text { if } x \leq 0} \\ {5} & {\text { if } 0 < x \leq 3 \text { . Compute each. }} \\ {-x^{2}} & {\text { otherwise }}\end{array}\right. $$

Step-by-Step Solution

Answer

To convert a Celsius temperature to Fahrenheit, use the formula: \(F = \frac{9}{5}C + 32\). For the piecewise function g(x), apply the appropriate rule based on the value of x. For instance, when computing g(-2), g(2), and g(5), the respective values are 7, 5, and -25.

Step 1: Converting Celsius to Fahrenheit

To convert a Celsius temperature to Fahrenheit, we use the formula: \(F = \frac{9}{5}C + 32\). We will need to substitute the given Celsius temperature into the formula and solve for the Fahrenheit temperature.

Step 2: Evaluating the piecewise function g(x)

The function g(x) has three different rules, depending on the value of x. We will need to determine which rule to apply based on the given value of x, and then use that rule to compute the value of g(x). Let's compute g(-2), g(2), and g(5) as examples: - For g(-2): Since -2 is less than or equal to 0, we use the rule \(2|x| + 3\). We substitute x with -2 and obtain: \(g(-2) = 2|-2|+3 = 2\cdot2+3 = 7\). - For g(2): Since 2 is greater than 0 and less than or equal to 3, we use the rule 5. Hence, \(g(2) = 5\). - For g(5): Since 5 is greater than 3, we use the rule \(-x^2\). We substitute x with 5 and obtain: \(g(5) = -5^2 = -25\). The respective values of g(-2), g(2), and g(5) are 7, 5, and -25.

Key Concepts

Temperature Conversion

When we talk about temperature conversion, we are referring to the process of changing a temperature reading from one scale to another. The two most familiar scales are Celsius and Fahrenheit. Celsius is commonly used in most of the world, while Fahrenheit is predominantly used in the United States.
It's important because it allows us to interpret temperatures correctly, depending on which measurement system we are familiar with. For instance:

• If you're used to Celsius, a day with a temperature of 30°C feels warm.
• However, in Fahrenheit, the same temperature would be 86°F.

These conversions are crucial in scientific calculations, travel, and when reading temperature-sensitive instructions that might be in a different unit than you're used to.

Celsius to Fahrenheit Formula

The formula to convert Celsius to Fahrenheit is one of the key equations you need to know for temperature conversions. The formula is:\[ F = \frac{9}{5}C + 32 \] where:

• \( F \) is the temperature in Fahrenheit
• \( C \) is the temperature in Celsius

To use the formula, follow these simple steps:
1. Multiply the Celsius temperature by \( \frac{9}{5} \). This increases the value to account for the different sizes of Celsius and Fahrenheit degrees.
2. Add 32 to the result from step 1. This accounts for the difference in the starting point of the two scales. Celsius starts at 0 for freezing, while Fahrenheit starts at 32.
For example, to convert 20°C to Fahrenheit:

• Multiply 20 by \( \frac{9}{5} \) to get 36.
• Add 32 to 36, yielding 68°F.

This formula is straightforward and essential for accurately converting temperatures across these two scales.

Evaluating Functions

Understanding how to evaluate functions, including piecewise functions, is an important skill in mathematics. A piecewise function uses different expressions for different intervals of input values.Let's learn through an example function given as \( g(x) \). The function is defined with different expressions based on the range of \( x \):

• \( 2|x| + 3 \) if \( x \leq 0 \)
• 5 if \( 0 < x \leq 3 \)
• \(-x^2\) otherwise

To evaluate a piecewise function:

• Determine the interval that your input value \( x \) falls into.
• Use the corresponding expression to calculate the result.

For example:

• For \( g(-2) \), since \(-2\) is less than or equal to 0, use \(2|x| + 3\). Calculating gives \( g(-2) = 7 \).
• For \( g(2) \), since \( 0 < 2 \leq 3 \), use the value 5. Hence, \( g(2) = 5 \).
• For \( g(5) \), since 5 is greater than 3, use \(-x^2\). This results in \( g(5) = -25 \).

Evaluating piecewise functions involves carefully checking the conditions and applying the right equation for the correct interval of \( x \).