Math Problem Statement

1. Diketahui f(x) = 1 - √(x - 3). Tentukan D(f) dan R(f)! 2. Diketahui f(x) = √(x + 1) dan g(x) = x^2 - 1. Tentukan: a. f ∘ g dan D(f ∘ g) b. g ∘ f dan D(g ∘ f) 3. Diketahui fungsi y = e^(-x^2). Tentukan D(f), R(f), dan gambar grafik fungsi tersebut!

Solution

Let's go through each question one by one.

Problem 1

Given: f(x)=1x3f(x) = 1 - \sqrt{x - 3} Determine the domain D(f)D(f) and the range R(f)R(f).

  1. Domain D(f)D(f): For f(x)=1x3f(x) = 1 - \sqrt{x - 3} to be defined, the expression inside the square root must be non-negative: x30x3x - 3 \geq 0 \Rightarrow x \geq 3 Therefore, the domain D(f)D(f) is: D(f)=[3,)D(f) = [3, \infty)

  2. Range R(f)R(f): Since x3\sqrt{x - 3} yields values from 0 to infinity, f(x)=1x3f(x) = 1 - \sqrt{x - 3} will yield values from 1 to negative infinity as xx increases. Thus, the range is: R(f)=(,1]R(f) = (-\infty, 1]

Problem 2

Given: f(x)=x+1andg(x)=x21f(x) = \sqrt{x + 1} \quad \text{and} \quad g(x) = x^2 - 1 Determine fgf \circ g, D(fg)D(f \circ g), gfg \circ f, and D(gf)D(g \circ f).

  1. fgf \circ g: The composition fg(x)=f(g(x))f \circ g(x) = f(g(x)). f(g(x))=f(x21)=(x21)+1=x2=xf(g(x)) = f(x^2 - 1) = \sqrt{(x^2 - 1) + 1} = \sqrt{x^2} = |x|

    So, fg(x)=xf \circ g(x) = |x|.

  2. Domain D(fg)D(f \circ g): Since fg(x)=xf \circ g(x) = |x| is defined for all real numbers, the domain D(fg)D(f \circ g) is: D(fg)=(,)D(f \circ g) = (-\infty, \infty)

  3. gfg \circ f: The composition gf(x)=g(f(x))g \circ f(x) = g(f(x)). g(f(x))=g(x+1)=(x+1)21=x+11=xg(f(x)) = g(\sqrt{x + 1}) = (\sqrt{x + 1})^2 - 1 = x + 1 - 1 = x

    So, gf(x)=xg \circ f(x) = x.

  4. Domain D(gf)D(g \circ f): For gf(x)=xg \circ f(x) = x, we need f(x)=x+1f(x) = \sqrt{x + 1} to be defined, which requires x+10x1x + 1 \geq 0 \Rightarrow x \geq -1. Thus, the domain D(gf)D(g \circ f) is: D(gf)=[1,)D(g \circ f) = [-1, \infty)

Problem 3

Given: y=ex2y = e^{-x^2} Determine the domain D(f)D(f), range R(f)R(f), and sketch the graph of the function.

  1. Domain D(f)D(f): Since ex2e^{-x^2} is defined for all real values of xx, the domain D(f)D(f) is: D(f)=(,)D(f) = (-\infty, \infty)

  2. Range R(f)R(f): Since ex2e^{-x^2} is an exponential function with a maximum value of 1 (when x=0x = 0) and approaches 0 as xx approaches ±\pm \infty, the range is: R(f)=(0,1]R(f) = (0, 1]

  3. Graph: The graph of y=ex2y = e^{-x^2} is a bell-shaped curve symmetric about the y-axis, peaking at y=1y = 1 when x=0x = 0 and gradually approaching y=0y = 0 as xx moves away from zero.

Would you like further details on any of these solutions or help with visualizing the graph?

Related Questions

  1. How do you determine the range of a square root function?
  2. Why is the domain for composite functions sometimes more restricted?
  3. What are typical methods for finding the range of exponential functions?
  4. How does the square root function affect the domain of compositions?
  5. What makes ex2e^{-x^2} different from typical exponential functions?

Tip

When determining the range of functions involving square roots or absolute values, carefully consider how these operations limit the possible output values.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Domain and Range
Exponential Functions
Square Root Function

Formulas

Domain condition for square root functions: for √(x - a), x ≥ a
Range of e^(-x^2): (0, 1]
Composition of functions: (f ∘ g)(x) = f(g(x))

Theorems

Domain and Range in Function Composition
Properties of Exponential Functions
Square Root Function Properties

Suitable Grade Level

Grades 11-12