Problem 54 - Algebra A Combined Function

Question

If \(f(x)=3 x+3, g(x)=4 x^{2}-6 x+3,\) and \(h(x)=5 x^{2}-7,\) find each function value. \(h(-2)\)

Step-by-Step Solution

Answer

The value is 13.

Step 1: Substitute the Value into the Function

To find the value of the function \(h(x)\) at \(x = -2\), we start by substituting \(-2\) into the function \(h(x) = 5x^2 - 7\). This gives us \(h(-2) = 5(-2)^2 - 7\).

Step 2: Calculate the Squared Term

Calculate \((-2)^2\). We know that \((-2)^2 = 4\). Substitute this value back into the equation, which results in \(h(-2) = 5 \times 4 - 7\).

Step 3: Multiply by the Coefficient

Multiply the squared term by the coefficient: \(5 \times 4 = 20\). So our equation becomes \(h(-2) = 20 - 7\).

Step 4: Perform the Final Subtraction

Subtract \(7\) from \(20\): \(20 - 7 = 13\). Thus, we find that \(h(-2) = 13\).

Key Concepts

Squaring Numbers

Squaring a number is one of the fundamental operations in mathematics. It simply means multiplying a number by itself. For example, squaring -2 means calculating

• i.e. \(-2 \times -2 \).
• When you square a negative number, the negative signs cancel out, as a negative times a negative is a positive.
• The result for \(-2^2 ext{or} (-2)^2 \) is 4, because i.e. (-2) \times (-2) = 4. Therefore, whenever you square any negative number, the result is always positive.

Understanding squaring is vital when dealing with quadratic expressions or equations. It's essentially the building block of more complex algebraic functions.

Substitution Method

The substitution method is a technique used to solve equations. In function evaluation, we use this method to find the function's value at a given point. The steps include:

• Taking the function given, such as i.e.

  Example

  given \(h(x) = 5x^2 - 7\).
• Then, replacing "\(x\)" with the specific value e.g. \(x = -2\).
• For this function, you'd substitute i.e.

  Result

  Replace i.e. \(-2\) for every occurrence of i.e. x, to get

  Formula:

  \(h(-2) = 5(-2)^2 - 7\).

By applying substitution, we simplify the process of evaluating the function at specific points, which is essential in both algebra and calculus.

Polynomial Functions

Polynomial functions are expressions involving terms with variables raised to whole number powers. They're written in the form: \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where \(a_n, a_{n-1},..., a_1, \text{and} \,a_0\) are coefficients, and \(n\) is a non-negative integer.

• In the given problem, i.e., the function \(h(x) = 5x^2 - 7\) is a polynomial of degree 2, known as a quadratic polynomial.
• Polynomial functions can have different degrees, and the degree influences the function's graph and properties.
• Quadratic polynomials typically have a parabolic graph and can have either a maximum or minimum point.

Polynomial functions are foundational in algebra and calculus, allowing for analysis of various phenomena such as motion, growth, and decline across numerous fields.