Problem 37 - Contemporary Precalculus

Question

If \(f(x)=x^{3}+c x^{2}+4 x-1\) for some constant \(c\) and \(f(1)=2,\) find \(c .[\text {Hint: Use the rule of } f \text { to compute } f(1) .]\)

Step-by-Step Solution

Answer

Answer: -2

Step 1: Substitute the given value x = 1 into the function f(x)

Substitute \(x = 1\) into the function \(f(x)=x^3 + cx^2 + 4x - 1\) to get: $$ f(1) = (1)^3 + c(1)^2 + 4(1) -1 $$

Step 2: Simplify the expression

Now, we simplify the expression obtained in step 1: $$ f(1) = 1 + c + 4 - 1 $$ $$ f(1) = c + 4 $$

Step 3: Use the given value for f(1) and solve for c

We know that \(f(1) = 2\). Now, we can replace f(1) with 2, and solve for c: $$ 2 = c + 4 $$ Rearrange to solve for c: $$ c = 2 - 4 $$ Finally, compute the value for c: $$ c = -2 $$ Thus, the constant \(c\) for the function is -2.

Key Concepts

Function Evaluation

Function evaluation is an important process in which we substitute specific values into a given function to find corresponding outputs. In this exercise, we are asked to evaluate the polynomial function defined as \( f(x) = x^3 + cx^2 + 4x - 1 \) at a specific point, which is \( x = 1 \). By substituting \( x = 1 \) into the function, we calculate \( f(1) \), moving through each term:

• The first term, \( x^3 \), becomes \( 1^3 = 1 \).
• The second term, \( cx^2 \), is \( c \times 1^2 = c \).
• The third term, \( 4x \), becomes \( 4 \times 1 = 4 \).
• The constant in the equation remains \(-1\).

This results in the expression \( f(1) = 1 + c + 4 - 1 \), simplifying to \( f(1) = c + 4 \). This process is straightforward and effective for understanding how variables affect the outcome of a function.

Equation Solving

To solve an equation, you must isolate the unknown variable. In this context, after evaluating the function at \( x = 1 \), we received an expression \( f(1) = c + 4 \). We know that the function value \( f(1) \) is given as \( 2 \). By setting \( c + 4 = 2 \), we form a simple linear equation. To find \( c \), perform the following steps:

• First, subtract 4 from both sides of the equation to isolate \( c \). This gives us \( c = 2 - 4 \).
• Proceed to simplify the result to find \( c = -2 \).

This straightforward approach to solving equations involves basic arithmetic operations such as subtraction and simplification, making it easy to find unknown constants like \( c \). It highlights the process of balancing an equation to maintain equality on both sides.

Constant Determination

Determining a constant in a function often involves plugging in known values and solving for the unknown. The constant \( c \) in our exercise must satisfy the condition \( f(1) = 2 \). Through function evaluation and solving the subsequent equation \( c + 4 = 2 \), we discern that \( c \) must be \(-2\). This step is crucial for several reasons:

• Firstly, it allows the function to pass through given points, maintaining the desired properties of the function.
• Secondly, constants greatly affect the shape and position of polynomial functions when plotted.

By determining \( c = -2 \), control is exercised over the polynomial's behavior at specific points, ensuring it aligns with given conditions. Understanding this concept proves helpful not only in equations but in assessing how individual components contribute to the overall graph of the function.