Problem 62 - Elementary and Intermediate Algebra

Question

Find \(h(5)\) and \(h(-2) .\) See Example 4. $$ h(x)=\frac{3}{x-4} $$

Step-by-Step Solution

Answer

\( h(5) = 3 \); \( h(-2) = -\frac{1}{2} \).

Step 1: Identify the Function

The function we are working with is \( h(x) = \frac{3}{x-4} \). We need to evaluate this function at specific values of \( x \).

Step 2: Calculate \( h(5) \)

Substitute \( x = 5 \) into the function: \[ h(5) = \frac{3}{5-4} = \frac{3}{1} = 3 \].

Step 3: Calculate \( h(-2) \)

Substitute \( x = -2 \) into the function: \[ h(-2) = \frac{3}{-2-4} = \frac{3}{-6} = -\frac{1}{2} \].

Step 4: Conclude with Results

Therefore, the values of the function at given points are: \( h(5) = 3 \) and \( h(-2) = -\frac{1}{2} \).

Key Concepts

Understanding Functions in Algebra

Functions in algebra are special kinds of rules that connect each input to exactly one output. This is like a machine where you put something in and get something specific out. In algebra, these inputs and outputs are usually numbers. Consider a function as a way to describe how numbers change from input, let's call it \( x \), to output, often represented as \( f(x) \) or \( h(x) \).

For example, if you have a function \( f(x) = x + 2 \), and you input 3, the machine adds 2 to 3, resulting in an output of 5. Functions help in predicting outcomes, analyzing relationships, and solving problems by understanding how variables interact.

Functions can be simple like linear ones where changes are constant, or more complicated with rules involving exponents or divisions, like the function \( h(x) = \frac{3}{x - 4} \) from our exercise.

The Substitution Method Explained

The substitution method is a straightforward technique used to find the outputs of a function for particular inputs. Imagine you have a function machine, and you want to know what comes out when you place a certain value into it.

Here's how you do it:

• Identify the function you are working with. For instance, \( h(x) = \frac{3}{x-4} \).
• Decide the value you wish to substitute, like substituting \( x = 5 \).
• Replace the input variable in the function equation with your chosen number. In our example, it’s plugging 5 into the place of \( x \), giving us \( h(5) = \frac{3}{5-4} \).
• Solve the equation to get the result: \( h(5) = 3 \).
  
  

By substituting different values of \( x \), you can quickly determine what the function outputs, giving you a better understanding of its behavior.

Exploring Rational Functions

Rational functions are a specific type of function found in algebra that involve ratios of polynomials. They look somewhat like fractions, hence the name 'rational', which stems from the word 'ratio'.

For the function \( h(x) = \frac{3}{x-4} \), we have a numerator, which is the number 3, and a denominator, \( x-4 \). The exciting part about rational functions is that their values depend significantly on what \( x \) is since the denominator cannot be zero. If it’s zero, the function becomes undefined.

When working with rational functions, keep in mind:

• The function is undefined wherever the denominator equals zero. For \( h(x) \), this happens at \( x = 4 \).
• Approaching undefined points, the function can show dramatic increases or decreases, leading it to 'blow up' to infinity or negative infinity.
• This characteristic makes rational functions unique and sometimes more challenging to grasp, yet fascinating as they often appear in real-world problem-solving scenarios.

Understanding these properties can help you manipulate and analyze rational functions in various algebraic contexts.