IB Mathematics Analysis and Approaches Topic SL…

The function $f$ is such that $f(3x - 2) = x^{3} + x$ for all $x \in R$ .

(a) Show that $f(x) = \left(\frac{x + 2}{3}\right)^{3} + \frac{x + 2}{3}$ .

(b) Show that $f$ is one-to-one.

(e) The graph of $y=f\left(\frac{e^3}{4}x^5\right)$ can be generated through a sequence of a vertical stretch and a vertical translation of the graph of . these two transformations, clearly stating the order in which they should be performed.

(e) Write down:

(i) the equation of the vertical asymptote of $y = f(x)$ ;

(ii) the equation of the horizontal asymptote of $y = f(x)$ ;

(iii) the equation of a line of symmetry of the graph of $y = f(x)$ .

(d) Find $(f \circ f)^{-1}(4)$ .

The function $f$ is defined by $f(x)=\ln(4x)$ , where $x>0$ .

The function $f$ is defined by $f(x) = \frac{2x + 1}{x - 3}$ , for $x \in \mathbb{R}$ , $x \neq 3$ .

The function $g$ is defined by $g(x) = \frac{1}{x + 1}$ , for $x \in \mathbb{R}$ , .

(a) Show that $(f \circ g)(x) = -\frac{x + 3}{3x + 2}$ .

(d) Find the domain and the range of $(f \circ g)^{-1}$ .

Let $f(x) = \dfrac{4x - 3}{x - 4}$ , for $x \neq 4$ .

(b) Find $f^{-1}(x)$ .

(d) Solve $f(x) = 5$ .

Consider the function $f(x) = \frac{2x^2 - 3x + 1}{x^2 - 4}$ .

(a) Find the domain of the function $f(x)$ .

(a) the exact value of $f\left(\frac{e^4}{4}\right)$ .

(b) the exact value of $f\left(\frac{1}{4}\right)$ .

(d) Using your result from part 3, or otherwise, the value of $f^{-1}(-2)$ .

(b) the two real values that must be excluded from the domain of $f \circ g$ .

(b) the graph of $f(x)$ , showing all asymptotes and intercepts.

IB Mathematics Analysis and Approaches SL 2.5—composite Functions, Identity, Finding Inverse Questionbank

IB Mathematics Analysis and Approaches SL 2.5—composite Functions, Identity, Finding Inverse Questionbank

IB Mathematics Analysis and Approaches SL 2.5—composite Functions, Identity, Finding Inverse Questionbank

IB Mathematics Analysis and Approaches SL 2.5—composite Functions, Identity, Finding Inverse Questionbank

IB Mathematics Analysis and Approaches (AA) Topic SL 2.5—composite Functions, Identity, Finding Inverse Questionbank