Ex 1.3 , 8 - Chapter 1 Class 12 Relation and Functions
Last updated at Jan. 28, 2020 by Teachoo
Last updated at Jan. 28, 2020 by Teachoo
Transcript
Ex 1.3, 8 Consider f: R+ → [4, ∞) given by f(x) = x2 + 4. Show that f is invertible with the inverse f−1 of given f by f-1 (y) = √(y−4) , where R+ is the set of all non-negative real numbers. f(x) = x2 + 4 f is invertible if f is one-one and onto Checking one-one f (x1) = (x1)2 + 4 f (x2) = (x2)2 + 4 Putting f (x1) = f (x2) (x1)2 + 4 = (x2)2 + 4 Rough One-one Steps: 1. Calculate f(x1) 2. Calculate f(x2) 3. Putting f(x1) = f(x2) we have to prove x1 = x2 (x1)2 = (x2)2 ⇒ x1 = x2 or x1 = –x2 Since f: R+ → [4,∞ ) So x ∈ R+ i.e. x is always positive, Hence x1 = –x2 is not true So, x1 = x2 ∴ f is one-one Check onto f(x) = x2 + 4 Let f(x) = y such that y ∈ [4,∞ ) y = x2 + 4 y – 4 = x2 x2 = y – 4 x = ±√(𝑦−4) Since f: R+ → [4,∞ ) x ∈ R+ , so x is positive Hence we cannot take x = −√(𝑦−4) x = √(𝑦−4) Since, y ∈ [4,∞ ) i.e. y is greater than or equal to 4 i.e. y ≥ 4 y – 4 ≥ 0 Hence the value inside root is positive Hence, √(𝑦−4) ≥ 0 x ≥ 0 Hence x is a real number which is greater than or equal to 0. ∴ x ∈ R+ Now, Checking for y = f(x) Putting value of x in f(x) f(x) = f(√(𝑦−4)) = (√(𝑦−4))^2+4 = y − 4 + 4 = y Thus, for every y ∈ [4,∞ ) , there exists x ∈ R+ such that f(x) = y Hence, f is onto Since f(x) is one-one and onto, So, f(x) is invertible And Inverse of x = 𝑓^(−1) (𝑦) = √(𝒚−𝟒)
Finding Inverse
Inverse of a function
How to check if function has inverse? Deleted for CBSE Board 2021 Exams only
Example 22 Deleted for CBSE Board 2021 Exams only
Ex 1.3, 5 Important Deleted for CBSE Board 2021 Exams only
How to find Inverse?
Example 28 Deleted for CBSE Board 2021 Exams only
Misc 11 Important Deleted for CBSE Board 2021 Exams only
Ex 1.3, 11 Deleted for CBSE Board 2021 Exams only
Example 27 Important Deleted for CBSE Board 2021 Exams only
Misc 1 Deleted for CBSE Board 2021 Exams only
Ex 1.3 , 6 Deleted for CBSE Board 2021 Exams only
Ex 1.3 , 14 Important Deleted for CBSE Board 2021 Exams only
Example 23 Important Deleted for CBSE Board 2021 Exams only
Misc 2 Deleted for CBSE Board 2021 Exams only
Ex 1.3 , 4 Deleted for CBSE Board 2021 Exams only
Example 24 Deleted for CBSE Board 2021 Exams only
Ex 1.3 , 8 Important Deleted for CBSE Board 2021 Exams only You are here
Example 25 Important Deleted for CBSE Board 2021 Exams only
Ex 1.3 , 9 Important Deleted for CBSE Board 2021 Exams only
Finding Inverse
About the Author