Problem 13 - 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 \(\mathrm{h}(x)\) $$ \mathrm{f}(x)=x^{2}, \mathrm{g}(x)=4+x $$

Step-by-Step Solution

Answer

a. \( h(x) = 4 + x^2 \); b. Domain: \( (-\infty, \infty) \); c. Range: \( [4, \infty) \); d. Graph: upward parabola.

Step 1: Find Expression for h(x)

Given \( h(x) = g(f(x)) \), we substitute the expressions for \( f(x) \) and \( g(x) \). First, calculate \( f(x) = x^2 \). Then, substitute this into \( g(x) \), which gives us \( g(f(x)) = 4 + f(x) \). This results in \( h(x) = 4 + x^2 \).

Step 2: Determine Domain of h(x)

The function \( f(x) = x^2 \) is defined for all real numbers as squaring any real number is valid. Additionally, \( g(x) = 4 + x \) is also defined for all real numbers. Hence, the domain of \( h(x) \), which is \( 4 + x^2 \), includes all real numbers. Thus, the domain of \( h(x) \) is \( (-\infty, \infty) \).

Step 3: Determine Range of h(x)

The range of \( f(x) = x^2 \) is \( [0, \infty) \) as squaring any real number gives a non-negative result. For \( g(x) = 4 + x \) where \( x = f(x) \), it adds 4 to this output, transforming the range to \( [4, \infty) \). Thus, the range of \( h(x) = 4 + x^2 \) is \( [4, \infty) \).

Step 4: Graph h(x)

To graph \( h(x) = 4 + x^2 \), start by noting it is a parabola opening upwards, shifted 4 units up from the origin due to the constant term +4. The vertex is at \((0, 4)\), and the graph is symmetric about the y-axis. The shape is identical to \( x^2 \) but translated upwards by 4 units.

Key Concepts

Domain

When we talk about the domain of a function, we are looking at the complete set of input values that the function can take. For our function, \( h(x) = 4 + x^2 \), we need to consider the domains of both \( f(x) = x^2 \) and \( g(x) = 4 + x \).

• The function \( f(x) = x^2 \) accepts any real number as input because any number squared is a valid mathematical operation. Hence, its domain is \( (-\infty, \infty) \).
• Likewise, \( g(x) = 4 + x \) also accepts all real numbers since you can add 4 to any real number. Thus, its domain is \( (-\infty, \infty) \).

Combining these, the domain of the composed function \( h(x) = 4 + x^2 \) is all real numbers, or \( (-\infty, \infty) \), as it inherits the domain from \( f(x) \) and \( g(x) \).
This means you can plug any real number into \( h(x) \), and it will produce a valid output.

Range

The range of a function refers to all possible output values. For \( h(x) = 4 + x^2 \), we determine this based on the behavior of \( f(x) = x^2 \) and how it affects \( g(x) = 4 + x \).

• Starting with \( f(x) = x^2 \), for any real input \( x \), the output is always a non-negative number, resulting in a range of \( [0, \infty) \). This is because squared numbers are always positive or zero.
• The function \( g(x) = 4 + x \) translates this output upwards by 4. So, the smallest value that \( g(x) \) can produce is 4 when \( x = 0 \), and it extends to infinity.

Therefore, the range of \( h(x) = 4 + x^2 \) starts from 4 and goes to infinity, expressed as \( [4, \infty) \). This shows that no matter the input, you can expect outputs of \( h(x) \) to always be 4 or greater.

Graphing Functions

Graphing \( h(x) = 4 + x^2 \) involves understanding how the function behaves visually on a coordinate plane.

• It is a transformed version of the basic parabola \( y = x^2 \). The term \(+4\) indicates a vertical shift upwards.
• The vertex of the graph is at the point \((0, 4)\) because \( h(x) \) is simply \( x^2 \) moved 4 units upwards.
• Because the graph is a parabola, it is symmetric about the y-axis and opens upwards.

Considering these points, drawing the graph includes plotting the vertex and understanding that for every unit you move horizontally away from \( x = 0 \), the value of \( h(x) \) grows quadratically. Two main points to remember for sketching: When \( x = 1 \) or \( x = -1 \), \( h(x) = 5 \), confirming the bell shape of the parabola shifted upwards.
This visualization helps to see why the range is \([4, \infty)\), as the graph starts at y = 4 and extends infinitely up.