Problem 12 - Precalculus

Question

Use the given function \(f\) to find and simplify the following: \- \(f(3)\) \- \(f(4 x)\) \- \(f(x-4)\) \- \(f(-1)\) \- \(4 f(x)\) -\(f(x)-4\) \- \(f\left(\frac{3}{2}\right)\) \- \(f(-x)\) \- \(f\left(x^{2}\right)\) $$f(x)=3-4 x$$

Step-by-Step Solution

Answer

1) -9, 2) 3-16x, 3) 19-4x, 4) 7, 5) 12-16x, 6) -1-4x, 7) -3, 8) 3+4x, 9) 3-4x^2.

Step 1: Find f(3)

To find \(f(3)\), substitute \(3\) into the function \(f(x)\). This gives us \(f(3) = 3 - 4\times3\). Calculate \(4\times3 = 12\). So, \(f(3) = 3 - 12 = -9\).

Step 2: Find f(4x)

Substitute \(4x\) into the function. This yields \(f(4x) = 3 - 4\times(4x)\). Simplify to get \(3 - 16x\). So, \(f(4x) = 3 - 16x\).

Step 3: Find f(x-4)

Substitute \(x-4\) into the function. This results in \(f(x-4) = 3 - 4\times(x-4)\). Distribute to get \(3 - 4x + 16 = 19 - 4x\). Thus, \(f(x-4) = 19 - 4x\).

Step 4: Find f(-1)

Substitute \(-1\) into the function. Then, \(f(-1) = 3 - 4\times(-1)\). This simplifies to \(3 + 4 = 7\). Hence, \(f(-1) = 7\).

Step 5: Find 4f(x)

Multiply the entire function by 4. So, \(4f(x) = 4\times(3 - 4x)\). Distribute to get \(12 - 16x\). Therefore, \(4f(x) = 12 - 16x\).

Step 6: Find f(x)-4

Subtract \(4\) from the function. Thus, \(f(x)-4 = (3 - 4x) - 4\). Simplify to get \(3 - 4 - 4x = -1 - 4x\). So, \(f(x) - 4 = -1 - 4x\).

Step 7: Find f(\frac{3}{2})

Substitute \(\frac{3}{2}\) into the function. Then, \(f\left(\frac{3}{2}\right) = 3 - 4\times\frac{3}{2}\). Calculate \(4\times\frac{3}{2} = 6\). So, \(f\left(\frac{3}{2}\right) = 3 - 6 = -3\).

Step 8: Find f(-x)

Substitute \(-x\) into the function. This results in \(f(-x) = 3 - 4\times(-x)\). Simplify to \(3 + 4x\). Thus, \(f(-x) = 3 + 4x\).

Step 9: Find f(x^2)

Substitute \(x^2\) into the function. Therefore, \(f(x^2) = 3 - 4\times x^2\). Simplify to \(3 - 4x^2\). Hence, \(f(x^2) = 3 - 4x^2\).

Key Concepts

Linear Functions

A linear function is a type of function that creates a straight line when graphed on a coordinate plane. The general form of a linear function is \(f(x) = mx + b\), where \(m\) represents the slope of the line and \(b\) represents the y-intercept. These functions are foundational in algebra and are characterized by their constant rate of change. In the exercise given, the function \(f(x) = 3 - 4x\) is a classic linear function. Here, \(-4\) is the slope, indicating that the function decreases by 4 units in the \(f(x)\) value for every 1 unit increase in \(x\). The number \(3\) is the y-intercept, showing where the line crosses the y-axis, i.e., when \(x = 0\), \(f(x) = 3\). Linear functions are simple yet powerful, appearing in many mathematical models.

Substitution Method

The substitution method involves replacing a variable in a function or equation with a given value. This method is commonly used to evaluate functions at specific points. For instance, to find \(f(3)\) in the exercise, you substitute \(3\) into the equation \(f(x) = 3 - 4x\), resulting in \(f(3) = 3 - 4 \cdot 3\). By calculating, you find \(f(3) = -9\). Similarly, for \(f(4x)\), you replace \(x\) with \(4x\), leading to \(f(4x) = 3 - 4 \times 4x = 3 - 16x\).

This method is not just for numbers but also for expressions, such as \(f(x-4)\) or \(f(-x)\). Substitution is a critical technique in algebra to simplify and solve equations efficiently.

Polynomial Simplification

Polynomial simplification involves reducing expressions to their simplest form, making them easier to work with and understand. When you substitute values or expressions into a polynomial, performing arithmetic operations can often leave you with a more complex polynomial than you started with.

For example, when finding \(f(x-4)\), you start with \(f(x) = 3 - 4(x-4)\). It involves distributing \(-4\) across the expression \(x-4\), resulting in \(-4x + 16\), and then combining it with the constant \(3\). Ultimately, this simplification yields \(19 - 4x\). Simplifying polynomials is essential because it leads to more manageable forms of expressions that are easier to interpret and solve.

Algebraic Expressions

Algebraic expressions are mathematical phrases containing numbers, variables, and operation symbols. These expressions form the basis of solving equations and understanding functions. In the exercise, terms like \(4f(x)\) and \(f(x)-4\) involve algebraic manipulation of expressions involving the function \(f(x) = 3 - 4x\).

• To find \(4f(x)\), you multiply the entire function by 4, leading to \(12 - 16x\).
• For \(f(x)-4\), you subtract 4 from the function, simplifying it to \(-1 - 4x\).

Understanding how to work with these expressions allows you to explore deeper into algebra, meshing operations like addition, subtraction, multiplication, and division with equations to form a fundamental understanding of mathematics.