Verified Answer :

  • The function \(f(x) = 3x - 4\) for \(x \geq 1\) is a linear ray starting at the point \((1, -1)\) and increasing with a slope of 3.
  • Since the function is strictly increasing, the range is determined by the minimum value at the lower bound of the domain \(x = 1\), which is \(f(1) = -1\).
  • The final range of the function is \(y \geq -1\).
  • Final Answer: \(\boxed{y \geq -1}\)

Explanation

  1. Analyzing the Function and its Domain

To analyze the function \(f(x) = 3x - 4\) with the restricted domain \(x \geq 1\), we first identify the properties of this linear function. The function is in the form \(y = mx + b\), where the slope \(m = 3\) and the \(y\)-intercept would be \((0, -4)\) if the domain included zero. Because the slope is positive (\(m = 3 > 0\)), the function is strictly increasing, meaning that as \(x\) increases, \(f(x)\) also increases. The domain is given as \(x \in [1, \infty)\).

  1. Describing the Graph

The graph of the function \(f(x) = 3x - 4\) for \(x \geq 1\) is a ray. To find the starting point of this ray, we evaluate the function at the lower bound of the domain, \(x = 1\):
\[f(1) = 3(1) - 4 = 3 - 4 = -1.\]
Thus, the graph starts at the point \((1, -1)\). Since the slope is 3, for every unit increase in \(x\), the value of \(y\) increases by 3. The graph extends infinitely into the first quadrant, starting from the point \((1, -1)\) and moving upwards to the right.

  1. Determining the Range of the Function

The range of a function is the set of all possible output values \(y\). Since the function \(f(x) = 3x - 4\) is strictly increasing, its minimum value occurs at the minimum value of its domain, \(x = 1\). We calculated that \(f(1) = -1\). As \(x\) approaches infinity, \(f(x)\) also approaches infinity. Therefore, the function takes on all values starting from \(-1\) and increasing without bound. This corresponds to the range \(y \geq -1\).

Examples

Linear functions are widely used to model constant rate changes in the real world. For example, consider a taxi service that charges a flat fee of R\$2.00 (base fare) plus R\$3.00 for every kilometer traveled. If you represent the cost \(C\) as a function of distance \(d\), you get \(C(d) = 3d + 2\). If the taxi has a minimum trip requirement of 1 km, the function is \(C(d) = 3d + 2\) for \(d \geq 1\). Determining the range of this function helps you understand the minimum cost of any trip (which would be \(3(1) + 2 = \boxed{R\$5}\)). This demonstrates how domain restrictions and range calculations help in practical financial planning.

Related Questions