Theory of Intelligence
11TH•Mathematics•Functions And Graphs•Exercise 2.1

Introduction

Practice set 2.1 for Functions and Graphs (function evaluation, domain/range, one-to-one/onto).

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quizQuestion 1 of 1
Q4

Question 4

Find the domain and the range of the function gg defined below:

Select Part:Part (i) of 5
(i)

g(x)=5−xg(x)=5-x

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Step-by-Step Solution

Question 4 • Part (i)

verifiedVerified Answer

(i) g(x)=5−xg(x) = 5 - x

Given:

g(x)=5−xg(x) = 5 - x

  • Domain (DgD_g): Any real number can be substituted for xx:

    • g(0)=5−0=5g(0) = 5 - 0 = 5
    • g(2)=5−2=3g(2) = 5 - 2 = 3
    • g(−3)=5−(−3)=8g(-3) = 5 - (-3) = 8 No division by zero and no negative square roots:
    Dg=R=(−∞,∞)\boxed{D_g = \mathbb{R} = (-\infty, \infty)}

  • Range (RgR_g): Let y=5−x  ⟺  x=5−yy = 5 - x \iff x = 5 - y. Any real output can be obtained by choosing x=5−yx = 5 - y:

    • For y=10  ⟹  x=5−10=−5  ⟹  g(−5)=10y = 10 \implies x = 5 - 10 = -5 \implies g(-5) = 10
    • For y=0  ⟹  x=5−0=5  ⟹  g(5)=0y = 0 \implies x = 5 - 0 = 5 \implies g(5) = 0
    Rg=R=(−∞,∞)\boxed{R_g = \mathbb{R} = (-\infty, \infty)}
Dg=R,Rg=R\boxed{D_g = \mathbb{R}, \quad R_g = \mathbb{R}}
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Number Line Lab•Interactive

Real Number Line & Domain/Range Projection

linear_scaleInteractive Number Lines & Evaluation Lab

Real Number Line: Domain & Range of g(x) = 5 - x

Continuous linear polynomial extending infinitely in both directions without restriction.

D = ℝ = (-∞, ∞)•R = ℝ = (-∞, ∞)
Presets:
-15Scrub x along number line+15
Evaluation:g(0) = 5
check_circleIn Real Domain (Valid)
infoY-Intercept (x = 0)

At x = 0, g(0) = 5 - 0 = 5. On the graph, this is the y-intercept (0, 5).

straighten1D Continuous Real Number Lines

Domain Axis (Inputs: x)D = ℝ = (-∞, ∞)
-∞+∞-15-10-5051015x = 0
Range Axis (Outputs: g(x) or y)R = ℝ = (-∞, ∞)
-∞+∞-15-10-5051015g(0) = 5

show_chart2D Cartesian Coordinate Plane

xy(0, 5)(5, 0)(0, 5)
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