Evaluating Functions - Basic Introduction | Algebra

Summary

This educational video script offers a comprehensive guide to evaluating functions and solving equations involving them. It begins with a simple example of evaluating a function f(x) = x^2 + 5x - 7 at x = 3, demonstrating the process of substitution and calculation. The script then progresses to more complex functions involving multiple variables, like f(x, y) = 2x^2 - 3xy + 4y^2, and illustrates how to find function values given certain inputs. It also tackles the challenge of finding variable values when the function's output is known, using algebraic manipulation and factoring. The video concludes with a problem-solving approach to quadratic equations, either through factoring or utilizing the quadratic formula, providing viewers with a robust understanding of function evaluation and equation solving.

Takeaways

• 📘 To evaluate a function, substitute the given value for the variable in the function's expression.
• 🔢 For functions like f(x) = x^2 + 5x - 7, replace x with the specified value and perform the arithmetic to find f(x).
• 📐 When dealing with functions of two variables, such as f(x, y) = 2x^2 - 3xy + 4y^2, substitute both variables with their given values to evaluate the function.
• 🔍 To find the value of a variable when the function's output is known, set the function equal to the output and solve the resulting equation.
• 📉 For quadratic equations, factoring can be used to find the variable's value, especially when the equation is in the form ax^2 + bx + c = d.
• 🔑 Factoring involves finding two numbers that multiply to the product of the leading coefficient and the constant term and add up to the middle coefficient.
• ✅ Verification of solutions is crucial; substitute the found values back into the original function to ensure they yield the expected output.
• 🔄 When factoring is challenging, the quadratic formula y = (-b ± √(b^2 - 4ac)) / (2a) can be used as an alternative method to find the variable's value.
• 📝 In functions with multiple variables, identify which variable's value is being sought and isolate it by substituting known values and solving the resulting equation.
• 🧮 Algebraic manipulation, including expanding, factoring, and simplifying expressions, is key to solving for variables in functions.

Q & A

What is the value of f(x) = x^2 + 5x - 7 when x is 3?

To find the value of f(x) when x is 3, substitute x with 3 in the function: f(3) = 3^2 + 5*3 - 7. This evaluates to 9 + 15 - 7, which equals 17.

How do you evaluate the function f(x, y) = 2x^2 - 3xy + 4y^2 for x = 2 and y = 3?

Substitute x with 2 and y with 3 in the function: f(2, 3) = 2*(2^2) - 3*(2*3) + 4*(3^2). This simplifies to 2*4 - 18 + 4*9, which equals 8 - 18 + 36, resulting in 26.

If f(x) = x^2 - 4x + 9 and f(x) is 5, what is the value of x?

Set f(x) equal to 5: 5 = x^2 - 4x + 9. Rearrange to form a quadratic equation: x^2 - 4x + 4 = 0. Factor the quadratic to get (x - 2)^2 = 0, which gives x = 2.

What is the process to find the value of y in the function f(x, y) = 3x^2 - 4xy + 5y^2 when f(4, y) = 100?

Replace x with 4 and set f(4, y) to 100: 100 = 3*(4^2) - 4*(4*y) + 5*y^2. Simplify and rearrange to form a quadratic in y: 5y^2 - 16y - 52 = 0. Factor or use the quadratic formula to find y = -2 or y = 26/5.

How do you factor a trinomial with a leading coefficient other than one, as seen in the function f(x, y) = 3x^2 - 4xy + 5y^2?

For the trinomial 5y^2 - 16y - 52, find two numbers that multiply to 260 (5*52) and add to -16. These numbers are -26 and 10. Rewrite the middle term and factor by grouping to get (y + 2)(5y - 26).

What is the alternative method to factoring for solving a quadratic equation like 5y^2 - 16y - 52 = 0?

Use the quadratic formula y = [-b ± sqrt(b^2 - 4ac)] / (2a). For the equation 5y^2 - 16y - 52 = 0, substitute a = 5, b = -16, and c = -52 to find the values of y.

How does the quadratic formula help in solving for y in the equation 5y^2 - 16y - 52 = 0?

Apply the quadratic formula: y = [16 ± sqrt((-16)^2 - 4*5*(-52))] / (2*5). This results in y = [16 ± sqrt(256 + 2080)] / 10, which simplifies to y = [16 ± sqrt(2336)] / 10, giving y = 26/5 or y = -2.

What is the significance of setting each factor in a factored equation to zero when solving for variables?

Setting each factor to zero allows you to solve for the variable values that satisfy the equation. In the factored form (y + 2)(5y - 26) = 0, setting y + 2 = 0 and 5y - 26 = 0 gives the solutions y = -2 and y = 26/5.

Why is it important to check your work after solving an equation, as demonstrated in the script?

Checking your work by substituting the found values back into the original equation ensures the solution is correct. For x = 2 in x^2 - 4x + 9 = 5, substituting shows 2^2 - 4*2 + 9 = 5, confirming the solution is accurate.

How does understanding the process of evaluating functions and solving equations help in grasping algebraic concepts?

Evaluating functions and solving equations provide practical applications of algebraic concepts like substitution, factoring, and using the quadratic formula, which are essential for a strong foundation in algebra.