Problem 74 - Contemporary Precalculus

Question

Use the catalog of basic functions (page 170 ) and Section 3.4 to describe the graph of the given function. $$h(x)=\sqrt{x}-2$$

Step-by-Step Solution

Answer

Question: Identify the key features of the function $$h(x) = \sqrt{x} - 2$$, including its function family, domain, range, transformations, and intercepts. Answer: The function $$h(x) = \sqrt{x} - 2$$ belongs to the square root function family. Its domain is $$x \geq 0$$, and its range is $$h(x) \geq -2$$. The function is vertically shifted down by 2 units compared to the basic square root function $$f(x) = \sqrt{x}$$. The x-intercept is at the point $$(4, 0)$$ and the y-intercept is at the point $$(0, -2)$$.

Step 1: Identify the basic function family

The given function is $$h(x) = \sqrt{x} - 2$$. From the catalog of basic functions, we can see that this function belongs to the square root family, which has the basic function $$f(x) = \sqrt{x}$$.

Step 2: Determine the domain and range

The basic square root function, $$f(x) = \sqrt{x}$$, has a domain of $$x \geq 0$$, since we cannot take the square root of a negative number. In the given function, there are no additional restrictions on the domain, so the domain remains $$x \geq 0$$. For the range, the basic square root function has a range of $$f(x) \geq 0$$, since the square root of any non-negative number is non-negative. In the given function, the transformation $$-2$$ shifts the graph vertically down by 2 units. Therefore, the range of our function is $$h(x) \geq -2$$.

Step 3: Identify any transformations from the basic function

Our given function is $$h(x) = \sqrt{x} - 2$$. Comparing it to the basic function $$f(x) = \sqrt{x}$$, we can see that there is a vertical shift. The $$-2$$ term indicates that the graph of the function is shifted downward by 2 units.

Step 4: Find the intercepts

To find the x-intercept(s), we set $$h(x) = 0$$ and solve for $$x$$: $$0 = \sqrt{x} - 2$$ $$2 = \sqrt{x}$$ $$x = 2^2$$ $$x = 4$$ So the x-intercept is at the point $$(4, 0)$$. To find the y-intercept, we set $$x = 0$$ and solve for $$h(x)$$: $$h(0) = \sqrt{0} - 2$$ $$h(0) = -2$$ The y-intercept is at the point $$(0, -2)$$. In conclusion, the graph of the function $$h(x) = \sqrt{x} - 2$$ is a square root function with a domain of $$x \geq 0$$ and a range of $$h(x) \geq -2$$. It has been vertically shifted down by 2 units relative to the basic function $$f(x) = \sqrt{x}$$. The x-intercept is at the point $$(4, 0)$$, and the y-intercept is at the point $$(0, -2)$$.

Key Concepts

Square Root Functions

The square root function is a fundamental mathematical function that belongs to the family of radical functions. It is represented by the expression \( f(x) = \sqrt{x} \). This basic function has a unique shape, which is a gradual curve that starts at the origin (0,0) and extends to the right.

• The square root function only takes non-negative inputs, meaning you cannot find the square root of negative numbers in basic real-valued functions.
• The graph of a square root function is restricted to the first quadrant of the coordinate plane, which means it only exists where \( x \) is greater than or equal to zero.

Understanding the square root function helps in visualizing and analyzing its behavior when transformations are applied, such as shifts or stretches.

Domain and Range

The domain of a function is the set of all possible input values (\( x \)-values) for which the function is defined. For the square root function \( f(x) = \sqrt{x} \), the domain is all non-negative real numbers, expressed as \( x \geq 0 \). This is because you cannot take the square root of a negative number without using complex numbers.
On the other hand, the range of a function refers to all possible output values (\( y \)-values) the function can produce. The range of the basic square root function is \( y \geq 0 \), indicating that the square root of any non-negative number is also non-negative. When reflecting on the function \( h(x) = \sqrt{x} - 2 \):

• The domain remains \( x \geq 0 \) since the square root component is unaffected by transformations to the output.
• However, the range changes due to the vertical shift, resulting in \( y \geq -2 \) because every output of the basic function is decreased by 2 units.

Graph Transformations

Graph transformations alter the appearance of the original function on the coordinate plane. There are several kinds of transformations, including translations, reflections, stretches and compressions.For \( h(x) = \sqrt{x} - 2 \), the transformation present is a vertical translation:

• Vertical translations shift the graph up or down. The \(-2\) signifies that every point on the basic square root graph \( f(x) = \sqrt{x} \) is moved 2 units downward.
• This results in a new graph where the shape remains the same, but each point's \( y \)-coordinate is decreased by 2.

Transformations do not alter the fundamental nature of the square root function; they merely shift its position or adjust its scale on the graph.

Function Intercepts

Intercepts are crucial for understanding the points where a graph intersects the \( x \)-axis and \( y \)-axis. They are particularly useful in sketching and contextualizing the form of a graph.To find the **x-intercept** of \( h(x) = \sqrt{x} - 2 \):

• Set \( h(x) = 0 \) and solve for \( x \).
• This gives us \( 0 = \sqrt{x} - 2 \rightarrow \sqrt{x} = 2 \rightarrow x = 4 \).
• Thus, the x-intercept is at \((4, 0)\).

For the **y-intercept**:

• Set \( x = 0 \) and calculate \( h(0) \).
• This results in \( h(0) = \sqrt{0} - 2 = -2 \).
• Therefore, the y-intercept is at \((0, -2)\).

These intercepts provide key anchor points for sketching the overall graph and confirming its position on the coordinate plane.