Problem 49 - Algebra 2

Question

If \(f(x)=2 x+4, g(x)=x-1,\) and \(h(x)=x^{2},\) find each value. $$ h[f(-3)] $$

Step-by-Step Solution

Answer

The value is 4.

Step 1: Evaluate f(-3)

First, we need to substitute into the function \(f(x) = 2x + 4\). This will give us the value of \(f(-3)\). Substitute \(-3\) for \(x\): \[f(-3) = 2(-3) + 4 = -6 + 4 = -2\].

Step 2: Substitute f(-3) into h(x)

Now that we have \(f(-3) = -2\), we substitute this value into \(h(x) = x^2\). Replace \(x\) with \(-2\) in \(h(x)\): \[h(-2) = (-2)^2 = 4\].

Key Concepts

Function Evaluation

Understanding function evaluation is like asking a machine to perform a specific task. Imagine a function as a recipe, and the variable is the ingredient you put in. Every time you have a specific value to plug into the function, you follow the steps it provides. Suppose you have a function \( f(x) = 2x + 4 \). If you're asked to evaluate \( f(-3) \), then everywhere you see an \( x \) in the recipe, you substitute \( -3 \). For our recipe, this means calculating \( 2(-3) + 4 \), which results in \( -6 + 4 = -2 \). Through function evaluation, you determine the output when you input a given value.

Substitution

Substitution in mathematics allows us to replace variables, like swapping puzzle pieces when you want to form a complete picture. When dealing with multiple functions, substitution is key. For our exercise, we first evaluated \( f(-3) \) to get \( -2 \). With substitution, this \( -2 \) now becomes the new value to work with. You then take this \( -2 \) and substitute it into another function, in this case, \( h(x) = x^2 \). In our puzzle, that means changing the \( x \) in \( h(x) \) to \( -2 \), leading us to calculate \( h(-2) = (-2)^2 = 4 \). This methodical replacement process helps simplify complex problems.

Quadratic Function

A quadratic function is a special type of function that forms a curve on a graph, known as a parabola. It's written in the form \( ax^2 + bx + c \). In our exercise, the function \( h(x) = x^2 \) is a simple quadratic function, where \( a = 1 \), and both \( b \) and \( c \) are zero. The beauty of quadratic functions is their symmetry and predictability. When they are evaluated with a specific number using substitution, as we did with \( -2 \) to find \( h(-2) = 4 \), they show how inputs relate to outputs in a way that's easy to graph. Understanding quadratics is crucial as they explain many real-world phenomena, like how objects fall under gravity.