There is an isosceles triangle with height 4cm. a. If the base is x cm, express the area S using x. b. If f(x) = S, draw the graph of f(x). c. Determine the domain and range of f(x... There is an isosceles triangle with height 4cm. a. If the base is x cm, express the area S using x. b. If f(x) = S, draw the graph of f(x). c. Determine the domain and range of f(x).
Understand the Problem
The question involves an isosceles triangle with a specified height. It requires expressing the area in terms of the base, drawing the graph of the area function, and determining the domain and range of that function.
Answer
The area is $S = 2x$, with $f(x) = 2x$, domain $(0, \infty)$, and range $(0, \infty)$.
Answer for screen readers
The area $S$ of the triangle is expressed as $S = 2x$. The function $f(x) = S = 2x$ can be graphed as a straight line with a slope of 2.
- Domain: $(0, \infty)$
- Range: $(0, \infty)$
Steps to Solve
- Express Area of the Triangle
The area $S$ of a triangle can be expressed using the formula:
$$ S = \frac{1}{2} \times \text{base} \times \text{height} $$
Given that the base is $x$ cm and the height is 4 cm, we substitute:
$$ S = \frac{1}{2} \times x \times 4 $$
This simplifies to:
$$ S = 2x $$
- Define the Function
We define the function $f(x)$ as equal to the area of the triangle:
$$ f(x) = S = 2x $$
- Graph the Function
To graph $f(x) = 2x$, note that it is a linear function with:
- A slope of 2.
- A y-intercept at (0, 0).
The line will pass through the origin and rise steeply as $x$ increases.
- Determine the Domain and Range
- The domain of $f(x)$ represents all possible values for the base $x$. Since a triangle must have a positive length for the base, we have:
$$ x > 0 \Rightarrow \text{Domain: } (0, \infty) $$
- The range of $f(x)$ will also start from 0 and increase without bound, as both $S$ and $x$ can grow indefinitely. Thus:
$$ \text{Range: } (0, \infty) $$
The area $S$ of the triangle is expressed as $S = 2x$. The function $f(x) = S = 2x$ can be graphed as a straight line with a slope of 2.
- Domain: $(0, \infty)$
- Range: $(0, \infty)$
More Information
The area of a triangle depends directly on its base and height. Here, we express it linearly in terms of the given base. Linear functions are straightforward and depict a consistent rate of change, evident in the graph of $f(x)$.
Tips
- Assuming $x$ can be negative or zero: Ensure that the base of the triangle is always positive.
- Miscomputing the area formula: Remember to correctly apply the area of a triangle formula by including both the height and base.
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