Question

Functions f and g are defined by f: x \mapsto 2x + 1, \quad x \in \mathbb{R}, x > 0, g: x \mapsto \frac{2x - 1}{x + 3}, \quad x \in \mathbb{R}, x \neq -3. (i) Solve the equation gf(x) = x. (ii) Express $f^{-1}(x)$ and $g^{-1}(x)$ in terms of x. (iii) Show that the equation $g^{-1}(x) = x$ has no solutions. (iv) Sketch in a single diagram the graphs of y = f(x) and y = f?¹(x), making clear the relationship between the graphs.

          Functions f and g are defined by
f: x \mapsto 2x + 1, \quad x \in \mathbb{R}, x > 0,
g: x \mapsto \frac{2x - 1}{x + 3}, \quad x \in \mathbb{R}, x \neq -3.
(i) Solve the equation gf(x) = x.
(ii) Express $f^{-1}(x)$ and $g^{-1}(x)$ in terms of x.
(iii) Show that the equation $g^{-1}(x) = x$ has no solutions.
(iv) Sketch in a single diagram the graphs of y = f(x) and y = f?¹(x), making clear the relationship between the graphs.

Functions f and g are defined by
f: x ↦2x + 1, x ∈ℝ, x > 0,
g: x ↦(2x - 1)/(x + 3), x ∈ℝ, x ≠-3.
(i) Solve the equation gf(x) = x.
(ii) Express f^-1(x) and g^-1(x) in terms of x.
(iii) Show that the equation g^-1(x) = x has no solutions.
(iv) Sketch in a single diagram the graphs of y = f(x) and y = f?¹(x), making clear the relationship between the graphs.

Added by Alan A.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Donna Densmore

05/05/2022

Step 1

However, the given equation does not provide any information about the functions f() and g(). Without knowing the specific forms of these functions, it is not possible to solve the equation. (ii) The expression ( (1) 3 pue '1JO SILIDI W! (r) is not clear and does Show more…

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A solution is defined by "+47 Hr:J "434 "0<r ~Iz 4i:3 TeR: **3 {+r Solve the equation gf() = x (ii) Express ( (1) 3 pue '1JO SILIDI W! (r) (iii) Show that the equation g ( = sey solutions. [tl (iv) Sketch in a single diagram the graphs of y = f(1) and y = f'(). making clear the relationship between the graphs.