Functions revision P1 [98 marks]
1.
[Maximum mark: 5]
The functions f and g are defined such that f (x) = x+3
and g (x) = 8x + 5.
4
2.
(a)
Show that (g ∘ f ) (x) = 2x + 11.
(b)
Given that (g ∘ f )
−1
(a) = 4, find the value of a.
[2]
[3]
[Maximum mark: 5]
The graph of y = f (x) for −4 ≤ x ≤ 6 is shown in the following diagram.
(a.i)
Write down the value of f (2).
[1]
(a.ii)
Write down the value of (f ∘ f )(2).
[1]
(b)
Let g(x) = 12 f (x) + 1 for −4 ≤ x ≤ 6. On the axes above, sketch
the graph of g.
[3]
3.
[Maximum mark: 5]
The following table shows values of f (x) and g(x) for different values of x.
Both f and g are one-to-one functions.
4.
(a)
Find g(0).
[1]
(b)
Find (f ∘ g)(0).
[2]
(c)
Find the value of x such that f (x) = 0.
[2]
[Maximum mark: 7]
Let g (x) = x2 + bx + 11. The point (−1, 8) lies on the graph of g.
(a)
Find the value of b.
(b)
The graph of f (x) = x2 is transformed to obtain the graph of g.
Describe this transformation.
5.
[3]
[4]
[Maximum mark: 5]
Consider the functions f (x) = x + 2 and g(x) = x2 − k2, where k is a real constant.
6.
(a)
Write down an expression for (g ∘ f ) (x).
[2]
(b)
Given that (g ∘ f )(4) = 11 , find the possible values of k.
[3]
[Maximum mark: 16]
(a)
The graph of a quadratic function f has its vertex at the point (3, 2) and
it intersects the x-axis at x = 5. Find f in the form
f (x) = a(x − h)
2
+ k.
[3]
The quadratic function g is defined by g(x) = px2 + (t − 1)x − p where x ∈ R
and p, t ∈ R, p ≠ 0.
In the case where g(−3) = g(1) = 4,
(b.i)
find the value of p and the value of t.
[4]
(b.ii)
find the range of g.
[3]
(c)
The linear function j is defined by j(x) = −x + 3p where x ∈ R and
p ∈ R, p ≠ 0.
Show that the graphs of j(x) = −x + 3p and
2
g(x) = px
+ (t − 1)x − p have two distinct points of intersection
for every possible value of p and t.
7.
[6]
[Maximum mark: 7]
7x+7
The function f is defined by f (x) = 2x−4 for x ∈ R, x ≠ 2.
8.
(a)
Find the zero of f (x).
[2]
(b)
For the graph of y = f (x), write down the equation of
(b.i)
the vertical asymptote;
[1]
(b.ii)
the horizontal asymptote.
[1]
(c)
Find f −1(x), the inverse function of f (x).
[3]
[Maximum mark: 6]
A function f is defined by f (x) =
2x−1
x+1
, where x ∈ R, x ≠ −1.
The graph of y = f (x) has a vertical asymptote and a horizontal asymptote.
(a.i)
Write down the equation of the vertical asymptote.
[1]
(a.ii)
Write down the equation of the horizontal asymptote.
[1]
(b)
On the set of axes below, sketch the graph of y = f (x).
On your sketch, clearly indicate the asymptotes and the position of any
points of intersection with the axes.
[3]
(c)
9.
Hence, solve the inequality 0 <
2x−1
x+1
< 2.
[Maximum mark: 8]
2(x+3)
A function f is defined by f (x) = 3(x+2) , where x ∈ R, x ≠ −2.
The graph y = f (x) is shown below.
[1]
(a)
Write down the equation of the horizontal asymptote.
[1]
Consider g(x) = mx + 1, where m ∈ R, m ≠ 0.
(b.i)
Write down the number of solutions to f (x) = g(x) for m > 0.
(b.ii)
Determine the value of m such that f (x) = g(x) has only one solution
for x.
(b.iii) Determine the range of values for m, where f (x) = g(x) has two
solutions for x ≥ 0.
10.
[Maximum mark: 5]
1
A function f is defined by f (x) = 1 − x−2
, where x ∈ R, x ≠ 2.
(a)
The graph of y = f (x) has a vertical asymptote and a horizontal
asymptote.
Write down the equation of
[1]
[4]
[2]
(a.i)
the vertical asymptote;
[1]
(a.ii)
the horizontal asymptote.
[1]
(b)
Find the coordinates of the point where the graph of y = f (x) intersects
(b.i)
the y-axis;
[1]
(b.ii)
the x-axis.
[1]
(c)
On the following set of axes, sketch the graph of y = f (x), showing all
the features found in parts (a) and (b).
[1]
11.
[Maximum mark: 15]
Consider the function f (x) = ax where x, a ∈ R and x > 0, a > 1.
2
The graph of f contains the point ( 3 , 4).
(a)
Show that a = 8.
[2]
(b)
Write down an expression for f −1(x).
[1]
Find the value of f −1(√32).
[3]
(c)
Consider the arithmetic sequence log8 27 ,
p > 1 and q > 1.
(d.i)
(d.ii)
12.
log8 p ,
log8 q ,
log8 125 , where
Show that 27, p, q and 125 are four consecutive terms in a geometric
sequence.
[4]
Find the value of p and the value of q.
[5]
[Maximum mark: 14]
The following diagram shows the graph of y = −1 − √x + 3 for x ≥ −3.
(a)
Describe a sequence of transformations that transforms the graph of
y = √ x for x ≥ 0 to the graph of y = −1 − √ x + 3 for x ≥ −3.
[3]
A function f is defined by f (x) = −1 − √x + 3 for x ≥ −3.
(b)
State the range of f .
[1]
(c)
Find an expression for f −1(x), stating its domain.
[5]
(d)
Find the coordinates of the point(s) where the graphs of y = f (x) and
y = f
(x) intersect.
−1
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