Problem 11 - Algebra 2 and Trigonometry

Question

In \(11-18 :\) a. Find \(h(x)\) when \(h(x)=g(f(x)) .\) b. What is the domain of \(h(x) ?\) c. What is the range of \(\mathrm{h}(x) ?\) d. Graph \(\mathrm{h}(x)\) $$ \mathrm{f}(x)=2 x+1, \mathrm{g}(x)=4 x $$

Step-by-Step Solution

Answer

a. \( h(x) = 8x + 4 \); b. Domain is \( \mathbb{R} \); c. Range is \( \mathbb{R} \); d. Graph is a straight line with intercept 4 and slope 8.

Step 1: Find Functions Composition

To find \( h(x) = g(f(x)) \), start by substituting \( f(x) \) into the function \( g(x) \). We have \( f(x) = 2x + 1 \), so substitute it into \( g(x) = 4x \):\[ h(x) = g(f(x)) = g(2x + 1) = 4(2x + 1) \]Calculate the expression to find \( h(x) \):\[ h(x) = 8x + 4 \]

Step 2: Determine Domain of h(x)

The domain of a function is the set of all possible input values \( x \). Since both \( f(x) \) and \( g(x) \) are linear and defined for all real numbers, \( h(x) \) is also a linear function and is defined for all real numbers. Thus, the domain of \( h(x) = 8x + 4 \) is all real numbers \( \mathbb{R} \).

Step 3: Determine Range of h(x)

The range of a function is the set of all possible output values. Since \( h(x) = 8x + 4 \) is a linear function with no restrictions on \( x \), it can take any real number as an output. Therefore, the range of \( h(x) \) is also all real numbers \( \mathbb{R} \).

Step 4: Graph h(x)

To graph \( h(x) = 8x + 4 \), start by identifying the y-intercept and the slope. The y-intercept is 4, which is where the line crosses the y-axis. The slope is 8, meaning for every unit increase in \( x \), \( h(x) \) increases by 8 units. Plot the y-intercept (0,4) and use the slope to plot another point (1,12), then draw the line through these points.

Key Concepts

Linear Functions

Linear functions are one of the simplest types of functions and are essential building blocks in algebra. A linear function has the general form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This form implies a straight line when you graph the function. It's called 'linear' because it forms a straight line when plotted on a coordinate plane.

Key characteristics of linear functions:

• They exhibit constant rates of change, which are represented by the slope \( m \).
• The slope \( m \) determines how steep the line is — a higher absolute value of \( m \) indicates a steeper line.
• The y-intercept \( b \) is the point where the line intersects the y-axis.
• They have a linear graph with no curves, bends, or deviations.

In our example, both \( f(x) = 2x+1 \) and \( g(x) = 4x \) are linear functions, with slopes 2 and 4, and a y-intercept of 1 for \( f(x) \). Combined into \( h(x) = 8x + 4 \), we get one single linear function, maintaining simplicity in form and behavior.

Domain and Range

Understanding the domain and range of a function is crucial in identifying the set of possible inputs (domain) and the possible outputs (range). A linear function typically has a domain and a range of all real numbers, unless specifically restricted.

For our function \( h(x) = 8x + 4 \):

• Domain: Linear functions like \( h(x) \) don't have any restrictions on \( x \), they accept any real number. Therefore, the domain of \( h(x) \) is all real numbers \( \mathbb{R} \).
• Range: Similarly, because there are no horizontal asymptotes or bounds to \( h(x) \), the function outputs any real number. Thus, the range is also all real numbers \( \mathbb{R} \).

Examining the domain and range helps us understand that \( h(x) \) can stretch across the entire coordinate plane, proving its linear nature.

Graphing Functions

Graphing a function helps visualize its behavior and overall properties. With linear functions, this process is straightforward due to their linearity.

To graph \( h(x) = 8x + 4 \), start with the key components:

• Y-intercept: This is located at 4, or point (0,4). It shows where the graph crosses the y-axis.
• Slope: The slope of 8 signals that for every increase of one unit in \( x \), \( h(x) \) increases by 8 units.

Using these points, plot the y-intercept and then move 8 units up (since the slope is positive) and 1 unit to the right to find another point on the line, such as (1,12). Draw the line through these points to visualize \( h(x) \).

Graphing not only shows the linearity but also highlights how drastic changes occur per unit increment of \( x \). For linear functions, such graphs are crucial for identifying slope steepness and intercepts.