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

IB Mathematics Analysis and Approaches Topic SL 2.5—composite Functions, Identity, Finding Inverse Questionbank (SL/HL) | RevisionDojo

Practice IB Mathematics Analysis and Approaches (AA) Topic SL 2.5—composite Functions, Identity, Finding Inverse with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for SL 2.5—composite Functions, Identity, Finding Inverse and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

SLPaper 1

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

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

[2]

Calculate the exact value of $f (\frac{1}{4})$ .

[1]

Determine an expression for the inverse function $f^{- 1} (x)$ .

[4]

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

[1]

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

[6]

Question 2

SLPaper 1

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

Find $f (0)$ .

[1]

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

[4]

Show that $(f \circ f) (x) = x$ , and state what this implies about the function $f$ .

[3]

Solve $f (x) = 5$ .

[3]

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

[3]

Question 3

HLPaper 2

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

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

[4]

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

[3]

Find the inverse of the function $f (x)$ over the restricted domain $[1, 2) \cup (2, 3]$ if it exists.

[6]

Question 4

HLPaper 1

Let $h (x) = \frac{1}{x ^{2}}$ and $k (x) = ln (x)$ .

Write down the composite function $(h \circ k) (x)$ .

[1]

Determine the domain of $(h \circ k) (x)$ . Exclude any $x$ for which $(h \circ k) (x) = \frac{1}{4}$ .

[5]

Evaluate the limit $lim_{x \to 1} (h \circ k) (x)$ .

[2]

Now consider the function $f (x) = k (x) h (x)$ . Find $\int f (x) d x$ .

[5]

Question 5

HLPaper 2

The function $g$ is defined by $g (x) = \frac{2 l n x + 1}{x - 3}$ , $0 < x < 3$ .

Sketch the graph of $y = g (x)$ , showing any axis intercepts and giving the equations of any asymptotes.

[4]

Hence, or otherwise, find the coordinates of the point of inflexion on the graph of $y = g (x)$ .

[4]

Sketch the graph of $y = g^{- 1} (x)$ , showing any axis intercepts and giving the equations of any asymptotes.

[4]

Hence, or otherwise, solve the inequality $g (x) > g^{- 1} (x)$ .

[4]

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

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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

Unlock all 10 questions with a free account