In the present paper, we study the shifted hypergeometric function
f
(
z
)
=
z
2
F
1
(
a
,
b
;
c
;
z
)
f(z)=z_{2}F_{1}(a,b;c;z)
for real parameters with
0
>
a
≤
b
≤
c
0>a\le b\le c
and its variant
g
(
z
)
=
z
2
F
2
(
a
,
b
;
c
;
z
2
)
g(z)=z_{2}F_{2}(a,b;c;z^2)
. Our first purpose is to solve the range problems for
f
f
and
g
g
posed by Ponnusamy and Vuorinen [Rocky Mountain J. Math. 31 (2001), pp. 327–353]. Ruscheweyh, Salinas and Sugawa [Israel J. Math. 171 (2009), pp. 285–304] developed the theory of universal prestarlike functions on the slit domain
C
∖
[
1
,
+
∞
)
\mathbb {C}\setminus [1,+\infty )
and showed universal starlikeness of
f
f
under some assumptions on the parameters. However, there has been no systematic study of universal convexity of the shifted hypergeometric functions except for the case
b
=
1
b=1
. Our second purpose is to show universal convexity of
f
f
under certain conditions on the parameters.