Problem 49 - 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. \(h(-3)\)

Step-by-Step Solution

Answer

The value of \(h(-3)\) is 38.

Step 1: Substitute -3 into h(x)

To find the value of the function \(h(x)\) at \(x = -3\), we start by substituting \(-3\) into the function \(h(x)\). The function is defined as \(h(x) = 5x^2 - 7\). So, substitute \(-3\) in place of \(x\) to get: \[h(-3) = 5(-3)^2 - 7\].

Step 2: Calculate the square of -3

The next step is to calculate \((-3)^2\). The result is \((-3) \times (-3) = 9\). Substitute this result back into the equation to get \[h(-3) = 5(9) - 7\].

Step 3: Multiply by 5

Now, multiply \(9\) by \(5\) to get \[5 \times 9 = 45\]. Then update the equation to \[h(-3) = 45 - 7\].

Step 4: Subtract 7

Finally, subtract \(7\) from \(45\) to get \[45 - 7 = 38\]. This gives us the value of the function \(h(-3)\).

Key Concepts

Substitution in Function Evaluation

When evaluating a function for a specific value, we use a mathematical technique known as substitution. Substitution involves replacing a variable, usually denoted as \(x\), with a given number. In our exercise, for instance, we substituted \(-3\) into the function \(h(x) = 5x^2 - 7\). This method simplifies the problem, transforming an expression that depends on a variable into one that is easier to manage: an expression with numbers only.

• Identify the variable and function: Here, it's \(x\) and \(h(x) = 5x^2 - 7\).
• Substitute the given value into the function: Replace \(x\) with \(-3\) to form \(h(-3) = 5(-3)^2 - 7\).
• Compute the resulting expression: Work through the arithmetic to find the result.

Understanding substitution is crucial for evaluating polynomial functions and many other types of mathematical expressions. It allows us to determine the output of a function for specific inputs, as seen in this exercise.

Polynomials and Their Characteristics

Polynomials are algebraic expressions that consist of variables and coefficients, using operations like addition, subtraction, and multiplication. These expressions can be as simple as a single term or as complex as many terms combined. In this exercise, the function \(h(x) = 5x^2 - 7\) is a polynomial.

• Terms: A polynomial like \(5x^2 - 7\) includes individual parts called terms. Here, \(5x^2\) and \(-7\) are terms.
• Degree: The degree of a polynomial is determined by the highest power of the variable. \(5x^2 - 7\) has a degree of 2 because of the term \(5x^2\).

Polynomials are essential because they often model real-world situations and are foundational in algebra. The key is to apply basic arithmetic rules to simplify and evaluate them, which we've done through substitution and calculation in the given exercise.

Understanding Quadratic Functions

Quadratic functions are a type of polynomial characterized by the square of the variable. They are expressed in the standard form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). In the exercise, \(h(x) = 5x^2 - 7\) is a quadratic function, since it fits this structure.

• Standard Form: The function is already presented as \(5x^2 + 0x - 7\), highlighting its quadratic nature.
• Parabolic Shape: Quadratic functions graph into a parabola, which can open upwards or downwards based on the leading coefficient (\(a\)).
• Vertex and Axis of Symmetry: Important features of quadratics that help in understanding their graphs, though not directly involved in our simple computation here.

Evaluating quadratic functions frequently involves simple operations like squaring and arithmetic, as demonstrated through substitution with \(x = -3\) in the problem. They are a cornerstone of algebra due to their prevalence in various mathematical and scientific applications.