A1.3and1.5-Functions

Suppose that f(x) has a domain of [9,15] and a range of [4,11]. What are the domain and range of: (a) f(x)+2 Domain: Range: (b) f(x+2) Domain: Range: (c) f(2x) Domain: Range: (d) 2f(x) Domain: Range:

(a) f(x)+2 is obtained by shifting f(x) upwards by 2 units. Therefore the domain remains [9,15] while the range becomes [6,13]. (b) f(x+2) is obtained by shifting f(x) by 2 units left along the x axis. Therefore the domain becomes [7,13] while the range remains [4,11]. (c) f(2x) is obtained by compressing f(x) by a factor of 2. Therefore the domain becomes [92,152] while the range remains [4,11]. (d) 2f(x) is obtained by stretching f(x) vertically by a factor of 2. Therefore the domain remains [9,15] while the range becomes [8,22].

Find the inverse f ⁻¹ of f(x)=(x−3)/(1+2x) (Use symbolic notation and fractions where needed.) f⁻¹(x)=

(x+3)/(1−2x)

Which of the following describes the domain of the function g(z)=(z+z⁻¹)/[(z+5)(z−4)]:

All z in R except z = 0, -5, 4.

Let f(x)=x⁵+x+8. Find the value of the inverse function at a point. (Use symbolic notation and fractions where needed.) f⁻¹(254)=

f(3)=3⁵+3+8=254; therefore, f⁻¹(254)=3.

Find the range of the functions f(x)=√(1²−x²), g(t)=sin(7/t), h(z)=1−|z−2|, expressing your answer in interval notation. f(x): g(t): h(z): Hint: What are the possible output values √x, sint, |z| ?

f(x):[0,1], g(t):[−1,1], h(z):(−∞,1].

What is the largest interval containing zero on which f(x)=sin x is one-to-one? lower bound: upper bound:

Looking at the graph of sin x, the function is one-to-one on the interval [−π/2,π/2].

Calculate the composite functions f∘g and g∘f. f(x)=cos(x), g(x)=6x³+9x²−4

Solution: f(g(x))= f(6x³9x²4)= cos(6x³+9x²−4); g(f(x))= g(cos(x))= 6cos³(x)+9cos²(x)−4.

Let f(x)=(x+9)²+9. Find the largest value of a so that f is one to one on the interval (−∞,a]. a= Consider f on the domain the interval (−∞,a] and let g be the inverse of the function f. Give the domain and the range of the function g. The domain is the interval: and the range is the interval: . Give a formula for the function g. g(x)=

The graph of f is a parabola with vertex at the point −9,9. The parabola will be one to on the interval to the left of −9 or the interval (−∞,−9]. The range of the function f is the interval [9,∞) since the square takes on all positive values. The domain of f is the range of the inverse function g and the domain of f is the range of inverse function g. Thus the domain of g is the interval [9,∞) and the range of g is the interval (−∞,−9]. To find a formula for g(x), we must solve y=(x+9)²+9 to express x in terms of y. Switching x and y in this equation then yields g(x)=−9-√(x−9)

Let f(x)=1/(x⁴+1), g(x)=x⁻⁴. Calculate the composite functions f(g(x)) and g(f(x)) and determine their domains. (Use symbolic notation and fractions where needed.) f(g(x))= The domain is: A. x>0 B. all real numbers C. x≠0 D. x≥0 g(f(x))= The domain is: A. x>0 B. x≠0 C. all real numbers D. x≥0

f(g(x))=1/(x⁻¹⁶+1); D:x≠0. g(f(x))=(x⁴+1)⁴; D:R.

Calculate the composite functions f∘g and g∘f. f(x)=6^x, g(x)=x³ f(g(x))= g(f(x))=

f(g(x))=f(x³)=6x³; g(f(x))=g(6^x)=6^(³x).