Functions — Solutions

Full worked solutions, with method and final answer for every part.

Question 1

(a)fg(x) means f(g(x)): substitute g(x) = x² into f in place of x.

f(x²) = 2x² + 3

Question 2

(a)Write y = 2x + 3, swap x and y, then rearrange to make y the subject: x = 2y + 3, so y = (x − 3)/2.

f⁻¹(x) = (x − 3)/2

Question 3

(a)fg(x) means f(g(x)): substitute g(x) = x+4 into f in place of x.

3(x+4) − 2 = 3x + 10

Question 4

(a)gf(x) means g(f(x)): substitute f(x) = 3x−2 into g in place of x.

(3x−2) + 4 = 3x + 2

Question 5

(a)Write y = 3x − 2, swap x and y, and rearrange: x = 3y − 2, so y = (x+2)/3.

f⁻¹(x) = (x + 2)/3

Question 6

(a)Substitute x = 3.

3² + 1 = 10

Question 7

(a)Substitute x=5.

(5−1)/2 = 2

Question 8

(a)Substitute x=−2, taking care with the negative squared.

2(4) − 3 = 5

Question 9

(a)First find g(3)=9, then find f(9).

f(9) = 9+5 = 14

Question 10

(a)The domain is the complete set of possible input values (x-values) for which the function is defined.

The set of all valid input (x) values for the function

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