Problem 39 - Thinking Mathematically

Question

Evaluate each function at the given value of the variable. \(g(x)=-x^{2}+2\) a. \(g(4)\) b. \(g(-3)\)

Step-by-Step Solution

Answer

The result of the function evaluation is \(g(4) = -14\) and \(g(-3) = -7\).

Step 1: Substitute \(x = 4\) into the function

To evaluate the function \(g(x)\) at \(x=4\), you substitute 4 for \(x\) in the function, which gives: \(g(4) = -4^{2} + 2 = -16 + 2 = -14\)

Step 2: Substitute \(x = -3\) into the function

To evaluate the function \(g(x)\) at \(x=-3\), you substitute -3 for \(x\) in the function, which gives: \(g(-3) = -(-3)^{2} + 2 = -9 + 2 = -7\)

Key Concepts

Substitution

Substitution is a fundamental concept in algebra and mathematics in general. It involves replacing a variable in an expression or equation with a specific value to evaluate or solve it. This process is essential in function evaluation, as it allows us to determine the output of a function for particular inputs. In the context of our exercise, we are dealing with the function \( g(x) = -x^2 + 2 \). To evaluate this function at specific values, we substitute those values into the function for \( x \).

For example:

• Substituting \( x = 4 \) into \( g(x) \) gives \( g(4) = -(4)^2 + 2 \).
• Substituting \( x = -3 \) into \( g(x) \) gives \( g(-3) = -(-3)^2 + 2 \).

Substitution simplifies the mathematical expression, making it possible to perform operations needed to find the final answer. It’s like filling in the blanks with the proper numbers. Make sure to carry out any arithmetic carefully to avoid errors.

Quadratic Function

Quadratic functions are a type of polynomial function characterized by their highest degree of a term being two. These functions have the general form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are coefficients and \( a \) is non-zero. The function \( g(x) = -x^2 + 2 \) from our exercise is a simple example of a quadratic function because its highest power of \( x \) is two. This particular function has no \( x \) term or linear component (\( b = 0 \)), resulting in a parabolic curve opening downwards because the coefficient of \( x^2 \) is negative.

A quadratic function can produce:

• A single value for each input, depending on the value of \( x \).
• A parabola as its graph, which can open upwards or downwards based on the sign of \( a \).

In the function \( g(x) \), when substituting values for \( x \), we calculate a specific output by squaring the \( x \) value, multiplying by -1, and then adding 2. This process helps us visualize and understand how inputs into a quadratic function are transformed into outputs.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and arithmetic operations. They form the basis of any algebraic work and include quantities that change, known as variables, and constants, which are fixed values.

In the example of our exercise, the algebraic expression \( -x^2 + 2 \) is used within the function \( g(x) \).

• The term \( -x^2 \) tells us to take the square of \( x \) and then apply a negative coefficient, effectively reflecting the parabola downwards.
• The constant term \( +2 \) shifts the whole graph upward by two units.

When evaluating an algebraic expression like \( g(x) \) with a specific \( x \) value (e.g., \( 4 \) or \(-3 \)), it becomes a simple arithmetic task. Replace the variable \( x \) with the given number, then perform the operations step-by-step:
1. Square the number.
2. Apply the negative sign to the squared result.
3. Add the constant \( 2 \).

Understanding these components ensures a better grasp of how to manipulate and evaluate algebraic expressions, providing a deeper understanding of the interactions between variables and constants in mathematics.