Problem 38 - Contemporary Precalculus

Question

$$\text { If } f(x)=\frac{d x-5}{x-3} \text { and } f(4)=3, \text { find } d$$.

Step-by-Step Solution

Answer

Answer: The value of d is 2.

Step 1: Substitute x = 4 into the function f(x)#

We are given f(x) = $$\frac{d x - 5}{x - 3}$$ and f(4) = 3. Replace x with 4 in the equation: $$f(\text {4}) = \frac{d(\text {4})-5}{\text {4}-3}$$.

Step 2: Solve for d #

We have now: $$3 = \frac{d(\text {4})-5}{\text {4}-3}$$. To isolate d, follow these steps: 1. Multiply both sides of the equation by the denominator: $$3(\text {4}-3) = d(\text {4})-5$$. 2. Simplify the equation: $$3(\text {1}) = d(\text {4})-5 \implies 3 = 4d - 5$$. 3. Add 5 to both sides: $$3+5 = 4d - 5 + 5 \implies 8 = 4d$$. 4. Divide both sides by 4: $$\frac{8}{4} = \frac{4d}{4} \implies 2 = d$$.

Step 3: State the solution #

The value of d is 2.

Key Concepts

Algebraic Functions

Algebraic functions are mathematical expressions constructed through the use of algebraic operations, such as addition, subtraction, multiplication, division, and raising to a power on variables. In the context of the given exercise, the function presented is \( f(x)=\frac{dx-5}{x-3} \), which is a rational function, a type of algebraic function where one polynomial is divided by another.

It is important to understand that the behavior of algebraic functions can significantly differ based on the input value for the variable. For instance, the function in the exercise has a domain restriction: the value for \(x\) cannot be 3 because it would make the denominator zero, which is undefined in mathematics. Hence, when handling algebraic functions, it is key to identify any restrictions on the variables before proceeding with any form of evaluation or manipulation.

Function Evaluation

Function evaluation is the process of determining the output value of a function for a given input value. For the exercise in question, the evaluation involves substituting \( x=4 \) into the function \( f(x) \) and solving the expression. When you substitute the value into \( f(4) \), it's essential to perform each arithmetic operation carefully to ensure a correct output value.

For example, evaluating \( f(4) \) requires replacing \(x\) with 4: \( f(4)=\frac{d(4)-5}{4-3} \). This yields a new expression where \(d\) is the only variable left, allowing for its determination through further algebraic manipulations. Function evaluation, as shown in this step, is not only about substitution but also about simplifying and solving the resulting equation correctly.

Variable Isolation

Variable isolation is a technique used in algebra to solve for one variable in terms of others in an equation. It often involves a series of steps undertaken to get the variable of interest alone on one side of the equation. In our exercise, we seek to isolate the variable \(d\) so that we can find its value.

To achieve this, we began by multiplying both sides of our evaluated function by the denominator, effectively removing the fraction. Following this, we simplified the resulting expression and then added 5 to both sides of the equation to further isolate \(d\). The final step was to divide by 4 — the coefficient of \(d\) — to finally solve for \(d\).

Understanding the step-by-step process and the rationale behind each manipulation is crucial. It enables you to solve for any variable within complex equations systematically, which is a foundational skill in algebra that applies to a wide array of mathematical problems.