Problem 30 - Algebra 2

Question

Let \(g(x)=2 x\) and \(h(x)=x^{2}+4 .\) Evaluate each expression. $$ (h \circ h)(-4) $$

Step-by-Step Solution

Answer

Consequently, \((h \circ h)(-4)\) evaluates to 404.

Step 1: Evaluate \(h(-4)\)

Replace all instances of \(x\) in the function \(h(x) = x^{2} + 4\) with -4. So, \(h(-4) = (-4)^{2} + 4 = 16 + 4 = 20\).

Step 2: Evaluate \(h(h(-4))\)

Now, use the value we just calculated (20) as the input to the function \(h\) again. So, replace \(x\) in \(h(x)\) with 20, that gives us \(h(20) = 20^{2} + 4 = 400 + 4 = 404\).

Key Concepts

Understanding Evaluating Functions

Evaluating functions is like using a recipe, where you substitute ingredients with specific values to get a result. Here, the "ingredients" are the values you plug into the function. In mathematical terms, functions act like machines that take input values, known as arguments, and produce outputs based on a defined rule or algebraic expression.

• For instance, if you have a function named \( f(x) \) and you're asked to evaluate it at a specific value, such as \( x = a \), you simply substitute \( a \) into the function, replacing each instance of \( x \).
• This process transforms \( f(x) \) into a numeric result, known as \( f(a) \).

Evaluating functions is an essential skill that helps in applying mathematical operations and analyzing real-world problems. It lays the groundwork for understanding more complex topics like function composition.

Exploring Quadratic Functions

Quadratic functions are a type of polynomial function that are characterized by a specific form, which is generally written as \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). These functions graph as parabolas, which can open upwards or downwards depending on the sign of \( a \).

• In the given exercise, the function \( h(x) = x^2 + 4 \) is a quadratic function. Here, the coefficient \( a \) is 1 (since it is \( 1x^2 \)), \( b \) is 0 (as the linear term is absent), and \( c \) is 4.
• The graph of this function is a parabola that opens upwards and is shifted 4 units up on the y-axis, owing to the constant term \( c \).

Quadratic functions are vital in modeling situations in physics, engineering, and finance, where variables exhibit such curvilinear relationships.

Demystifying Function Notation

Function notation is a shorthand way of using symbols to describe the outputs of functions in relation to their inputs. It presents a clear and systematic method to specify the operations performed by functions.

• For example, the notation \( f(x) \) specifies a function named \( f \) with respect to the variable \( x \).
• When you see something like \( (h \circ h)(x) \), it refers to the composition of the function \( h \) with itself, meaning you evaluate the function with its own output as the new input.

Function notation simplifies communicating complex operations and is crucial in describing relationships between variables through mathematical expressions.