Problem 15 - Introductory Algebra for College Students

Question

Evaluate each function at the given values. \(g(x)=x^{2}+3 x\) a. \(g(2)\) b. \(g(-2)\) c. \(g(0)\)

Step-by-Step Solution

Answer

The evaluations are \(g(2) = 10\), \(g(-2) = -2\), and \(g(0) = 0\).

Step 1: Evaluate at \(g(2)\)

Substitute 2 for x in the function equation \(g(x) = x^{2} + 3x\). So, \(g(2) = 2^{2} + 3(2) = 4 + 6 = 10\)

Step 2: Evaluate at \(g(-2)\)

Substitute -2 for x in the function equation \(g(x) = x^{2} + 3x\). So, \(g(-2) = (-2)^{2} - 3(2) = 4 - 6 = -2\)

Step 3: Evaluate at \(g(0)\)

Substitute 0 for x in the function equation \(g(x) = x^{2} + 3x\). So, \(g(0) = 0^{2} + 3(0) = 0 + 0 = 0\).

Key Concepts

Algebraic Functions

An algebraic function, such as the one in our exercise, is composed of mathematical operations like addition, subtraction, multiplication, division, and exponentiation with variables. Essentially, these functions represent relationships between variables, usually with one variable depending on another.

For instance, in the function given by the equation \(g(x) = x^2 + 3x\), we see a polynomial, which is one of the most common types of algebraic functions. When evaluating algebraic functions, we replace the variable with the given input values and perform the operations as indicated. This is exactly what was done in the steps provided: for each value (2, -2, and 0), the value was substituted into the function's equation in place of \(x\), and the resulting expression was simplified to find the corresponding output.

Understanding algebraic functions is key to solving many mathematical problems, as they often represent a wide range of physical phenomena, from simple linear relationships to more complex patterns found in the natural and social sciences.

• Substitution is essential for evaluating functions at specific values
• Algebraic functions can be represented and analyzed graphically
• An understanding of algebra is fundamental to manipulating algebraic functions

Quadratic Functions

A quadratic function is a type of algebraic function characterized by the highest exponent of the variable being 2. The standard form of a quadratic function is \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not equal to zero.

The function \(g(x) = x^2 + 3x\) from our exercise is a quadratic function without a constant term (\(c = 0\)). Quadratic functions are known for their distinctive 'U'-shaped graphs called parabolas. When evaluating quadratic functions, just like other algebraic functions, we substitute the given values into the \(x\) variable and simplify the result.

To fully grasp quadratic functions, you must understand their:

• Graphs — parabolas will open upwards if \(a > 0\) and downwards if \(a < 0\)
• Vertex — the highest or lowest point on the graph
• Axis of symmetry — a vertical line that divides the parabola into two symmetrical halves
• Zeros or roots — values of \(x\) where the function's value is zero, intersecting the x-axis on the graph

Recognizing these aspects will improve your ability to analyze and graph quadratic functions effectively.

Function Notation

Function notation is a way to denote functions in a format that facilitates understanding which values are outputs as a result of specific inputs. It follows the format \(f(x)\), where \(f\) is the name of the function, and \(x\) is the input variable.

In our exercise, \(g(x)\) is used to denote the function, with \(g\) being the function's name and \(x\) as the variable input. When evaluating functions, you'll often see instructions like \(g(2)\), which means 'find the output of function \(g\) when the input is 2'.

By indicating functions with this particular notation, mathematicians efficiently convey which function they are referring to and what the independent variable is, without resorting to writing out the whole equation every time. Mastery of function notation allows for pronounced clarity and brevity in mathematical expressions and communications.

• 'Function of x' doesn't imply multiplication
• Function notation delineates variables from constants
• Function notation is integral for differentiating between multiple functions within the same context