Question

Challenge Problem Suppose \(f(x)=x^{2}-4 x+c\) and \(g(x)=\frac{f(x)}{3}-4 .\) Find \(f(3)\) if \(g(-2)=5\)

Short answer

f(3) = 12

Step-by-step solution

Express g(x) in terms of f(x)

Given the function definitions, express the function g(x) in terms of f(x). We have: \[ g(x) = \frac{f(x)}{3} - 4 \]

Substitute in g(-2)

According to the problem, \( g(-2) = 5 \). Substitute -2 into g(x) and set it equal to 5. \[ 5 = \frac{f(-2)}{3} - 4 \]

Solve for f(-2)

Rearrange the equation from Step 2 to solve for \( f(-2) \). \[ 5 + 4 = \frac{f(-2)}{3} \]This gives us: \[ 9 = \frac{f(-2)}{3} \]Multiplying both sides by 3: \[ f(-2) = 27 \]

Use the f(x) function to express f(-2) in terms of c

Using the function definition \( f(x) = x^2 - 4x + c \), substitute -2 for x. \[ f(-2) = (-2)^2 - 4(-2) + c \]We know \( f(-2) = 27 \), so: \[ 27 = 4 + 8 + c \]Simplify to find c: \[ 27 = 12 + c \]Thus, \[ c = 15 \]

Find f(3)

Now that we have determined \( c = 15 \), substitute 3 for x in the function \( f(x) \) using this value of c. \[ f(3) = 3^2 - 4(3) + 15 \]Simplify: \[ f(3) = 9 - 12 + 15 \]Thus, \[ f(3) = 12 \]

Key concepts

Quadratic Equations

In this exercise, we work with a quadratic function. A quadratic equation is any polynomial equation of the form: \[ f(x) = ax^2 + bx + c \], where a, b, and c are constants, and a is not equal to zero. Quadratics are critical in algebra because they describe parabolic shapes and are used in various practical applications, from physics to finance. In our problem, the given function is \[ f(x) = x^2 - 4x + c \]. This function’s behavior depends on the value of c, which we need to determine by using additional information provided in the problem. Understanding how to manipulate and solve quadratic equations is essential in algebra. The solutions of a quadratic equation are typically found using the quadratic formula, factoring, or completing the square, depending on the given form of the equation.

Function Operations

In mathematics, functions describe relationships between inputs and outputs. Function operations include addition, subtraction, multiplication, and division of functions. In our problem, we use the operation between two functions: - Given:\[ f(x) = x^2 - 4x + c \] - We have \[ g(x) = \frac{f(x)}{3} - 4 \],This form of function operation scales the function f(x) by dividing it by 3, then shifts it downward by 4 units. Operations on functions are essential as they show how combining different processes affects the results. Therefore, understanding these operations helps in solving function-related problems and modeling real-world scenarios.

Problem-Solving Steps

When faced with algebraic problems, breaking them down into clear steps makes them easier to tackle. Here’s a methodical approach used in solving the given exercise:

• Identify Function Relationships: Recognize the given functions and how they relate.
• Substitution: Substitute the given specific values into the functions.
• Solve Simple Equations: Rearrange and solve for the unknowns step-by-step.
• Back-Substitute: Use the found value(s) to further compute required outputs.

By following these steps, students can systematically approach complex problems. Each stage builds upon the previous one, ensuring that errors are minimized and solutions are easier to understand and verify.

Substitution Method

The substitution method is a technique used to solve equations by substituting one variable's equivalent from one equation into another. In our exercise, we applied this method effectively. Here's how:1. We had the function \[ g(x) = \frac{f(x)}{3} - 4 \]2. We substituted a specific value (\[ x = -2 \]) into g(x) and equated it to the given result (\[ g(-2) = 5 \]) 3. This allowed us to express \[ f(-2) \] in a simpler form. Solving this, we found \[ f(-2) = 27 \]4. Next, we used the definition of \[ f(x) \] to find the value of c by substituting \[ x = -2 \]: \[ f(-2) = (-2)^2 - 4(-2) + c = 27 \]5. Finally, knowing c, we substituted back into \[ f(x) \] to find \[ f(3) \]. The substitution method is powerful in breaking down complex relationships and finding unknowns systematically.