Problem 40

Question

$$ \frac{1}{x-3} ; a=1 $$

Step-by-Step Solution

Verified

Answer

The value of the function at a = 1 is -1/2.

1Step 1: Understand the Given Function

We are given the function $ f(x) = \frac{1}{x-3} $ and we need to evaluate or understand this function when $ a = 1 $. This means substituting $ x = 1 $ into the function.

2Step 2: Substitute the Given Value

Substitute $ x = 1 $ into the function: $ f(x) = \frac{1}{x - 3} $. This becomes $ f(1) = \frac{1}{1 - 3} $.

3Step 3: Simplify the Expression

Simplify the expression $ f(1) = \frac{1}{1 - 3} $. Perform the subtraction in the denominator: $ 1 - 3 = -2 $. Therefore, $ f(1) = \frac{1}{-2} $.

4Step 4: Write the Final Answer

The value of the function $ f(x) $ at $ a = 1 $ is $ f(1) = -\frac{1}{2} $. This gives us the function's value at this specific point.

Key Concepts

Evaluating functions at a pointSimplifying expressionsSubstitution in functions

Evaluating functions at a point

When we talk about evaluating functions at a point, we are essentially interested in finding the output value of a function for a particular input. A function, such as $ f(x) = \frac{1}{x-3} $, relies on its input $ x $ to produce an output. By specifying $ a = 1 $, you are instructed to calculate what the function yields when $ x = 1 $.

First, replace $ x $ with the given number, which is $ a = 1 $. This transforms the function to $ f(1) = \frac{1}{1-3} $.
The main goal is to compute the result of this transformation to find the particular value at point $ a $.

Computing the value at this specific input allows you to understand how the function behaves at that point. It's like checking the function's response to a specific signal.

Simplifying expressions

Simplifying expressions is like cleaning up math to make it more understandable. It means performing all possible arithmetic and reducing fractions when you substitute a number into a function. In this case, after inserting $ x = 1 $ into the function $ f(x) = \frac{1}{x-3} $, the expression becomes $ f(1) = \frac{1}{1-3} $.

Start with solving the subtraction in the denominator: $ 1 - 3 $.
Calculate $ 1 - 3 = -2 $.
The fraction $ \frac{1}{-2} $ simplifies to the consistent and final result of $ -\frac{1}{2} $.

Through steps like these, you make the function's output easily interpretable and concise.

Substitution in functions

Substitution in functions is a method where you replace the variable in a function with a particular value. This technique allows you to pinpoint exactly what the function equals at specific inputs. In this exercise, the function is $ f(x) = \frac{1}{x-3} $, and you're told to substitute $ x = 1 $.

Switch out the $ x $ with the value you're given, which turns the function into $ f(1) = \frac{1}{1-3} $.
The substitution is a direct swap inside the expression, facilitating the evaluation of the function at that point.

Doing this correctly paves the way for you to process and understand the essence of the function’s behavior at that specific input.

Problem 40

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