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1 5 2 Given that f: x → 2x+5, g: x → x² and h : x → , where x ER and x 0 or - x find the following composite functions. a fg b gh fh hf ff hh g gfh h hgf с d e f

1 5 2 Given that f: x → 2x+5, g: x → x² and h : x → , where x ER and x 0 or - x find the following composite functions. a fg b gh fh hf ff hh g gfh h hgf с d e f

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please help me answer 1C, 2B, 2D, 2H

### Exercise 2C 1. **Given Functions:** - \( f : x \mapsto 5 - x \) - \( g : x \mapsto \frac{4}{x} \) **Conditions:** \( x \in \mathbb{R} \) and \( x \neq 0 \) or 5 **Tasks:** Find the values of the following compositions. a. \( ff(7) \) b. \( ff(-19) \) c. \( gg(1) \) d. \( gg\left( \frac{1}{2} \right) \) e. \( gg \, gg\left( \frac{1}{2} \right) \) f. \( ffff(6) \) g. \( fg \, fg(2) \) h. \( fg \, gf(2) \) **Tip:** - To form a composite function, the domain of \( f \) must be chosen so that the entire range of \( f \) is included in the domain of \( g \). This exercise explores the composition of functions and requires understanding domain restrictions to properly form composite functions.
Transcribed Image Text:### Exercise 2C 1. **Given Functions:** - \( f : x \mapsto 5 - x \) - \( g : x \mapsto \frac{4}{x} \) **Conditions:** \( x \in \mathbb{R} \) and \( x \neq 0 \) or 5 **Tasks:** Find the values of the following compositions. a. \( ff(7) \) b. \( ff(-19) \) c. \( gg(1) \) d. \( gg\left( \frac{1}{2} \right) \) e. \( gg \, gg\left( \frac{1}{2} \right) \) f. \( ffff(6) \) g. \( fg \, fg(2) \) h. \( fg \, gf(2) \) **Tip:** - To form a composite function, the domain of \( f \) must be chosen so that the entire range of \( f \) is included in the domain of \( g \). This exercise explores the composition of functions and requires understanding domain restrictions to properly form composite functions.
### Composite Functions and Expressions #### Problem 2 Given functions: - \( f: x \mapsto 2x + 5 \) - \( g: x \mapsto x^2 \) - \( h: x \mapsto \frac{1}{x} \), where \( x \in \mathbb{R} \) and \( x \neq 0 \) or \( -\frac{5}{2} \) Find the following composite functions: - a. \( fg \) - b. \( gh \) - c. \( fh \) - d. \( hf \) - e. \( ff \) - f. \( hh \) - g. \( gfh \) - h. \( hgf \) #### Problem 3 Given functions: - \( f: x \mapsto \sin{x} \) - \( g: x \mapsto x^3 \) - \( h: x \mapsto x - 3 \), where \( x \in \mathbb{R} \) Find the following functions: - a. \( hf \) - b. \( fh \) - c. \( fhg \) - d. \( fg \) - e. \( hhh \) - f. \( gf \) #### Problem 4 Given functions: - \( f: x \mapsto x + 4 \) - \( g: x \mapsto 3x \) - \( h: x \mapsto x^2 \), where \( x \in \mathbb{R} \) Express each of the following in terms of \( f, g, h \) as appropriate: - a. \( x \mapsto x^2 + 4 \) - b. \( x \mapsto 3x + 4 \) - c. \( x \mapsto x^4 \) - d. \( x \mapsto 9x^2 \) - e. \( x \mapsto 3x + 12 \) - f. \( x \mapsto 3 \left( x^2 + 8 \right) \) - g. \( x \mapsto 3x + 16 \)
Transcribed Image Text:### Composite Functions and Expressions #### Problem 2 Given functions: - \( f: x \mapsto 2x + 5 \) - \( g: x \mapsto x^2 \) - \( h: x \mapsto \frac{1}{x} \), where \( x \in \mathbb{R} \) and \( x \neq 0 \) or \( -\frac{5}{2} \) Find the following composite functions: - a. \( fg \) - b. \( gh \) - c. \( fh \) - d. \( hf \) - e. \( ff \) - f. \( hh \) - g. \( gfh \) - h. \( hgf \) #### Problem 3 Given functions: - \( f: x \mapsto \sin{x} \) - \( g: x \mapsto x^3 \) - \( h: x \mapsto x - 3 \), where \( x \in \mathbb{R} \) Find the following functions: - a. \( hf \) - b. \( fh \) - c. \( fhg \) - d. \( fg \) - e. \( hhh \) - f. \( gf \) #### Problem 4 Given functions: - \( f: x \mapsto x + 4 \) - \( g: x \mapsto 3x \) - \( h: x \mapsto x^2 \), where \( x \in \mathbb{R} \) Express each of the following in terms of \( f, g, h \) as appropriate: - a. \( x \mapsto x^2 + 4 \) - b. \( x \mapsto 3x + 4 \) - c. \( x \mapsto x^4 \) - d. \( x \mapsto 9x^2 \) - e. \( x \mapsto 3x + 12 \) - f. \( x \mapsto 3 \left( x^2 + 8 \right) \) - g. \( x \mapsto 3x + 16 \)
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