Problem 21 - Algebra and Trigonometry

Question

\(17-22=\) Use \(f(x)=3 x-5\) and \(g(x)=2-x^{2}\) to evaluate the expression. $$ \begin{array}{ll}{\text { (a) }(f \circ g)(x)} & {\text { (b) }(g \circ f)(x)}\end{array} $$

Step-by-Step Solution

Answer

(f ∘ g)(x) = 1 - 3x^2; (g ∘ f)(x) = -9x^2 + 30x - 23.

Step 1: Understanding Composition of Functions

Composition of functions is an operation where one function is applied to the result of another function. If we have functions \(f(x)\) and \(g(x)\), then \((f \circ g)(x)\) means \(f(g(x))\), which means substitute \(g(x)\) inside \(f(x)\). Similarly, \((g \circ f)(x)\) means \(g(f(x))\).

Step 2: Evaluate (f ∘ g)(x)

First, substitute \(g(x)\) in \(f(x)\):\[f(g(x)) = f(2-x^{2})\]We know \(f(x) = 3x - 5\). Hence,\[f(2 - x^2) = 3(2 - x^2) - 5 = 6 - 3x^2 - 5 = 1 - 3x^2\].

Step 3: Evaluate (g ∘ f)(x)

Substitute \(f(x)\) in \(g(x)\):\[g(f(x)) = g(3x - 5)\]We know \(g(x) = 2 - x^2\). Hence,\[g(3x - 5) = 2 - (3x - 5)^2\]Expand \((3x - 5)^2\): \[(3x - 5)^2 = (3x)^2 - 2 \times 3x \times 5 + 5^2 = 9x^2 - 30x + 25\]Substitute back:\[g(3x - 5) = 2 - (9x^2 - 30x + 25) = 2 - 9x^2 + 30x - 25 = -9x^2 + 30x - 23\].

Key Concepts

Function Evaluation

Function evaluation is the process of determining the output of a given function for a specific input. It's like figuring out what happens when you "plug in" a number or an expression into the function. For example, if you have the function \( f(x) = 3x - 5 \), to evaluate \( f(2) \), substitute \( x \) with 2: \( f(2) = 3(2) - 5 = 1 \). Similarly, if the input is another function, like \( g(x) = 2 - x^2 \), the evaluation would involve plugging the entire function into \( f(x) \). This concept is pivotal in understanding how compositions work as it lays the foundation for using one function's output as another function's input, leading to new insights and results. The art of function evaluation enhances problem solving by allowing detailed analysis of interactions between different mathematical expressions.

Quadratic Function

Quadratic functions are a special type of function where the highest degree of the variable is 2. They have the general form \( g(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). In our example, \( g(x) = 2 - x^2 \) is a quadratic function with terms arranged as \( -x^2 + 0x + 2 \). Quadratics typically produce a parabolic graph that opens upwards if \( a > 0 \) and downwards if \( a < 0 \). Some key features of quadratic functions include:

• Vertex: the highest or lowest point on the graph.
• Axis of symmetry: a vertical line that divides the parabola into two mirror-image halves.
• Roots or zeros: the points where the graph intersects the x-axis.

Understanding quadratic functions is crucial because they often model real-world phenomena, from physics to finance, helping predict future trends and make informed decisions.

Function Substitution

Function substitution involves inserting one function into another to create a new function. Imagine you're playing a game of nesting boxes – each box (function) can be placed inside another. For functions \( f(x) \) and \( g(x) \), the notation \( (f \circ g)(x) \) means substitute \( g(x) \) into \( f(x) \), resulting in a new function expression. For example, if \( f(x) = 3x - 5 \) and \( g(x) = 2 - x^2 \), substituting \( g \) into \( f \) gives \( f(g(x)) = f(2 - x^2) = 1 - 3x^2 \).Similarly, \( (g \circ f)(x) \) means substitute \( f(x) \) into \( g(x) \). Here, you'd compute \( g(f(x)) = g(3x - 5) = -9x^2 + 30x - 23 \). This substitution creates opportunities to explore complex relationships and patterns within mathematics by synthesizing individual functions into a cohesive whole.