Problem 72 - Algebra 2

Question

Let \(f(x)=x^{2}+1\) and \(g(x)=3 x .\) Find each value. \((f \circ g)(-3)\)

Step-by-Step Solution

Answer

The value of \((f \circ g)(-3)\) is 82.

Step 1: Define Functions and Target Value

First, identify the given functions \(f(x) = x^{2}+1\) and \(g(x) = 3x\). The value to be found is \((f \circ g)(-3)\), which means \(f(g(-3))\).

Step 2: Input into Function g

Substitute the given value into function g first, \(g(-3) = 3(-3)\). Hence, \(g(-3)\) equals to -9.

Step 3: Input into Function f

The next step is substituting this result into function f. Therefore \(f(g(-3)) = f(-9) = (-9)^2 + 1\). Calculate to get the final result.

Key Concepts

Quadratic Functions

Quadratic functions are a fundamental concept in algebra and are essential for understanding more complex mathematical relationships. These functions are characterized by a standard form equation:

• The general formula for a quadratic function is \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \).

The term \( ax^2 \) makes the function a "quadratic." When graphed, quadratic functions will create a parabolic curve. This curve can open upwards or downwards depending on the sign of \( a \).

• A positive \( a \) results in a parabola that opens upwards, resembling a "U" shape.
• A negative \( a \) makes the parabola open downwards, resembling an "n" shape.

The vertex of the parabola is its highest or lowest point, making it a crucial feature for graphing and solving quadratic equations. For our exercise, the function \( f(x) = x^2 + 1 \) is a straightforward quadratic function, with \( a = 1 \), \( b = 0 \), and \( c = 1 \). This means our parabola opens upwards and has its vertex at (0,1). It simply shifts the basic parabolic shape up by one unit.

Linear Functions

Linear functions represent the simplest type of function you'll encounter in algebra and are crucial to understanding more complex functions. They are described by a formula:

• The general formula for a linear function is \( g(x) = mx + b \), where \( m \) and \( b \) are constants.

Here, \( m \) is the slope of the line, indicating its steepness, and \( b \) is the y-intercept, the point at which the line crosses the y-axis. The slope \( m \) tells you how much the function's output changes for a one-unit increase in \( x \).

• If \( m \) is positive, the line ascends from left to right.
• If \( m \) is negative, it descends from left to right.

The function \( g(x) = 3x \) given in our exercise is linear with \( m = 3 \) and \( b = 0 \). This indicates a line that passes through the origin (point (0,0)) and has a slope of 3, meaning it rises three units vertically for every one unit it moves horizontally. This leads to a straight line graph that is quite steep.

Function Evaluation

Function evaluation is the process of determining a function's output for a given input. This involves substituting the input value into the function's equation and simplifying, producing the output or result.In our exercise, we have two functions, \( f(x) = x^2 + 1 \) and \( g(x) = 3x \). The goal is to find \((f \circ g)(-3)\), which involves evaluating the composition of functions.To begin, substitute \(-3\) into the function \( g(x) \). Calculating \( g(-3) \) yields:

• \( g(-3) = 3(-3) = -9 \).

Now, take the result and input it into \( f(x) \). This means we need to calculate \( f(-9) \):

• \( f(-9) = (-9)^2 + 1 = 81 + 1 = 82 \).

This final result, 82, is the output of \( (f \circ g)(-3) \). Understanding function evaluation, especially in function composition, is essential for solving complex problems and is a skill utilized often in math and science.